Paleocosmology: Observational Constraints on
the Evolution of Galaxies and
Large-Scale Structure
by
Eric James Gaidos
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctorate of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1996
@ Massachusetts Institute of Technology 1996. All rights reserved.
0
Author.
Department of Physics
May 23, 1996
Certified by.
Claude R. Canizares
Professor
Thesis Supervisor
Accepted by ...
-ASSACHUSETTS INSTHU
OF TECHNOLOGY
JUN 0 5 1996
Paleocosmology:
Observational Constraints on the
Evolution of Galaxies and
Large-Scale Structure
by
Eric James Gaidos
Submitted to the Department of Physics
on May 23, 1996, in partial fulfillment of the
requirements for the degree of
Doctorate of Philosophy
ABSTRACT
The study of galaxies and their clustering is important to understanding their formation and evolution but also to place constraints on cosmological models. I present two
surveys and their analyses: The first is an optical CCD imaging survey of 67 Abell
clusters to determine the distribution of luminosities and morphologies of the constituent galaxies. A composite luminosity function constructed from the photometric
data is well described by a Schechter function with a faint-end slope a = 1.09 + 0.08
and characteristic absolute magnitude Mý = -21.13 ± 0.11 + 5 log h. A systematic
brightening of M* with higher cluster redshift and/or cluster density is observed. The
data suggests an excess over the best-fit Schechter LF at MR > -16.5. The form of
the LF found here is in good agreement with field surveys but conflicts with recent
claims for a steep LF in clusters. My photometry of Brightest Cluster Galaxies finds
a dispersion of 0.06 magnitudes about a secular trend with redshift. The luminosity
evolution is more consistent with models with a flat IMF (giant dominated), but a
larger photometric sample is needed to substantiate this. I distinguish between disk
and spheroidal galaxies using the moments of the surface brightness profiles and estimate the fraction of disk galaxies. The mean fraction is approximately 30% and shows
a significant increase in clusters with z > 0.2, consistent with the Butcher-Oemler
effect. The second survey is an on-going search for X-ray emitting galaxy clusters in
archival Einstein IPC images that includes a multi-aperture source detection routine.
An algorithm is developed to quantify the certainty of source extent and select X-ray
cluster candidates. The surface density of bright X-ray clusters on the sky is consistent with results from the Rosat All Sky Survey, although the numbers involved are
very small. I ]present an analysis of CCD imaging follow-up to an X-ray source that
appears to be a cluster at a redshift of about 0.3.
Thesis Supervisor: Claude R. Canizares
Title: Professor
ACKNOWLEDGMENTS
I wish to express my sincere thanks to Prof. Claude Canizares, my Thesis Committee
Chairman, as well as Prof. John Huchra and Prof. Samir Mathur for their assistance
as well as patience in the completion of this thesis. Profs. Charles Steidel and John
Tonry provided invaluable advice throughout the course of the project. Thanks go
to Prof. Paul S!chechter who was daring enough to give a young engineer a crack at
astronomy five years ago. The MDM Observatory Time Allocation Committee was
very generous in its allotment of telescope time to this thesis project. This project
also benefited enormously from the many software packages and databases maintained
on the Internet and the many kind people who assisted me in using them. Finally, a
most heart-felt thank you to family and friends who supported me both emotionally
and intellectually on the long and winding road to a Ph.D; this thesis is dedicated to
you.
"...And whether you perceive it or not,
no doubt the Universe is unfolding as it should..."
- Max Ehrmann, Desiderata
(.•'
f7
O&areA5p--AK
Contents
1
Galaxies: The Most Ancient of Fossils
1.1
Luminosities & Morphologies as Artifacts ...
1.2
...A nd as Tools
1.3
Techniques & Previous Research
1.4
2
12
..............
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1
Luminosities . ......
1.3.2
M orphologies
. ..................
.
.........
..........
Motivation for this Thesis
.
........
.........
..
18
19
19
........
25
........................
26
A Wide-Field Optical Survey of Cluster Galaxies
28
2.1
Description of the Survey and Galaxy Catalog . ............
29
2.1.1
Cluster selection
29
2.1.2
Observations
2.1.3
Image Processing ..................
2.1.4
Photometry .......
..
2.1.5
Im age Reduction
.........................
2.1.6
M agnitude Errors
2.1.7
Com pleteness ...........................
43
2.2
Field Galaxy Counts ...........................
45
2.3
Global Galaxy Cluster Properties . ..................
2.4
.........................
........
..
..
2.3.2
M orphology ............................
LF Construction
..........
32
..
..
.........
32
....
35
37
........................
Cluster Richness
2.4.1
.....
......
2.3.1
Luminosity Functions
..
.........................
..........................
.........................
43
.
51
51
55
58
58
2.5
2.6
3
2.4.2
Composite Luminosity Function
2.4.3
Variation of the Cluster Luminosity Function
2.4.4
Discussion
. . . . . . . . . . . . . . . .
61
..................
Brightest Cluster Galaxies . . . . . . . . . . . . . .
2.5.1
The M1 - M* Differential
2.5.2
Discussion ...
.....
. . . . . . . . . .
...........
Morphology of Cluster Galaxies . . . . . . . . . . .
2.6.1
The Technique
2.6.2
Application to the Data
................
. . . . . . . . . . .
Galaxy Clustering: The Cosmic Rosetta Stone?
100
3.1
Background ....................
100
3.2
Galaxy Cluster Surveys
3.3
X-ray Clusters ...................
104
3.3.1
X-ray Emission from Clusters . . . . . .
104
3.3.2
X-ray Cluster Surveys
. . . . . . . . . .
108
Limitations of X-ray Surveys
. . . . . . . . . .
111
3.4
. . . . . . . . . . . . .
102
4 An Archival EINSTEIN IPC X-ray Cluster Surve•y
4.1
4.2
The Source Catalog
. . . . . . . . . .
4.1.1
Einstein IPC
..........
4.1.2
Source Detection . . . . . . . .
115
4.1.3
Source Identification . . . . . .
117
4.1.4
Source Extent Test . . . . . . .
118
4.1.5
The EMSS Survey Revisited . .
123
Optical Counterparts
.
o.
.
.
o
.e
.
.
o.
o
.........
124
°
4.2.1
Digitize Sky Survey Imaging . .
124
4.2.2
CCD Imaging of Galaxy Cluster Counterparts
124
X-ray Cluster Candidates
4.3.1
4.4
115
115
.rt
........
4.3
114
Bright Sources
. . . . . . .
. . . . . . . . .
Sources with PE > 0.99 . . . . . . . . .
.
.
.o .
.
o.
o. .
.
o. .
.
.
129
. .
129
129
5
Look-Back Time
143
5.1
A Picture-Puzzle of Galaxies and Clusters
. ..............
143
5.2
Lessons Learned from My Field Work . .................
146
5.3
Looking Forwards .............................
147
A Eine Kleine Theory: Galaxy Formation
149
B Cluster Luminosity Functions
158
B.1 Photometric Luminosity Functions
. ..................
B.2 Non-Photometric Luminosity Functions ...........
158
. . . . . .
162
List of Figures
1
Field and Cluster LF - Schechter parameters
Survey redshifts and filters
. . . . . . . . . . .
.....................
*..
76
CPE reiection criteria and magnitude errors . . . . . . . . . . . . . .
76
Field galaxy counts
. .... ... .
77
. .... ....
78
. . . . . . . . . . . . . . .
78
..................
Effects of Galactic extinction
.............
Effect of seeing and detection filter width
Average star counts and background model parameter covariance
Cluster luminosity overdensity
. .
. . . . . . . . . . . . . . . . . . . . .
Cluster luminosity overdensity and peak index
79
80
. . . . . . . . . . . .
80
. . . . . . . . . . . . . . . . . .
81
. . . . . . . . . . . . . . . . . . . . .
82
Error covariance and LF from outer galaxies . . . . . . . . . . . . . .
83
Schechter LF parameter for clusters
. . . . . . . . . . . . . . . . . .
83
M* dependence on cluster morphology
. . . . . . . . . . . . . . . . .
84
M* dependence on cluster overdensity
. . . . . . . . . . . . . . . . .
85
. ... .... .
86
Distibution of BCG and non-BCG clusters . . . . . . . . . . . . . . .
87
Concentration index vs. peak index
Composite luminosity function
BCG magnitudes ....................
. . . . . ....
88
M, - M* vs. cluster morphology indices . . . . . . . . . . . . . . . .
89
M, vs. cluster morphologies ..............
. .... ... .
90
Hubble Atlas of galaxy morphologies by Q parameter . . . . . . . . .
91
Image degradation for test of Q parameter . . . . . . . . . . . . . . .
92
M1-
M* vs. M1-
M2
............
Test of Q parameter with image degradation
Fraction of disk galaxies vs. redshift
. . . .
Fraction of disk galaxies vs. cluster morphology
Fraction of disk galaxies vs. morphology indices
IPC Source Counts ...........
. . . . . . . . . . . . . . . . . 131
X-ray Source Identification
. . . . . . . . . . . . . . . . . . . . . . . 132
Flux ratios of point sources
. . . . . . . . . . . . . . . . . . . . . . . 133
Signal-to-noise ratios of point sources
. . . . . . . . . . . . . . . . . 134
Signal-to-noise ratios of Abell clusters
. . . . . . . . . . . . . . . . . 135
Extent Probability & EMSS Recovery
. . . . . . . . . . . . . . . . . 136
EMSS/Archival IPC comparison
. . . . . . . . . . . . . . . . . . . . 136
X-ray Extent of EMSS Clusters ....
. . . . . . . . . . . . . . . . . 137
POSS E images of Abell Clusters . . . . . . . . . . . . . . . . . . . . 138
UK Schmidt J Images of Abell Clusters
. . . . . . . . . . . . . . . . 139
Gunn Filters and Elliptical Colors . . . ... ... ... .. .... .. 140
CCD Photometry Tests
13
........
. . . . . . . . . . . . . . . . . 140
EX0806.3+2057 Imaging ........
141
List of Tables
1
Abell Cluster Galaxy Survey
..............
2
Cluster Morphology Indices
.....................
3
Cluster Morphology Indices (cont.)
4
Background Fields
5
Variation of LF with Redshift & Cluster Properties . .........
1
Point sources with PE > 0.99
2
Unidentified Extended X-ray Sources . .................
..
. ..................
............................
......................
.....
96
..
97
98
99
99
132
142
Chapter 1
Galaxies: The Most Ancient of Fossils
The discipline of extragalactic astronomy resembles paleontology and anthropology
in that one of its principle concerns is the study of the evolution of populations
over periods of time vastly exceeding the human life span.
And like those other
sciences, it relies on the existence of two kinds of "fossil records" to overcome this
obstacle: The first is the ability to observe distant objects directly as they once were
(in astronomy, due to the finite speed of light) and compare these with objects at
the present epoch which one presumes to be their descendents. The second record is
the set of those characteristics of present objects which have preserved information
about their formation and evolution. And in a similar fashion these records have
their limitations: In the former it is necessary to connect the distant, young objects
with nearby, older counterparts; in the latter one must infer past events from limited
information in the present.
Galaxies are the observable and innumerable denizens of the Cosmos. They
are the gravitationally bound objects in which most of the star formation in the
Universe takes place, where heavy-element nucleosynthesis occurs, and where the
metal-enriched gas is converted into interesting and complex things like planets and
Life and astronomers. They have also existed for most of the estimated age of the
Universe; from nuclear chronology and stellar age estimates, our Galaxy is at least
10 Gyr old, and some of its associated globular cluster system at least 12 Gyr old
(Scully & Olive 1995; Bolte 1992).
The masses and luminosities of galaxies extend over a range of nearly five
orders of magnitude: A galaxy such as our own Milky Way has a mass in excess of
5 x 1011 solar masses (Little & Tremaine 1987) and shines with the light of nearly
1010 Suns or 3.8 x 1043 ergs sec - 1 (Binney & Tremaine 1987). The mean space density
of galaxies is about one per cubic mega-parsec (Mpc 3 ). With photographic plates on
the 100 inch Mount Wilson telescope, Edwin Hubble was able to detect galaxies to
an apparent red magnitude of about 17 or an equivalent flux level of 2.7 x 10- 12 ergs
sec-1 cm
-2
and distances of about 300 Mpc. Now, with a little effort (and a 10 meter
telescope equipped with solid-state detectors on a Hawaiian mountain-top), one can
observe galaxies more than an order of magnitude more distant and at flux levels
lower by four orders of magnitude. At this apparent magnitude limit, the density of
galaxies on the sky is 7 x 10i deg - 2 and, in principle, 3 x 1010 galaxies are observable
(Smail et al. 1995). By observing to cosmologically significant distances and lookback times approaching their estimated ages, we observe galaxies as they were in their
youth or even birth (Steidel et al. 1996). The numbers of brighter galaxies for which
more detailed information such as redshifts and morphologies can be has obtained has
also grown. Surveys of nearby galaxies now routinely include thousands of objects
(Loveday et al. 1992; Marzke et al. 1994; Ellis et al. 1995; Santiago et al. 1996; Lin
et al. 1996). With such large numbers one can describe galaxies in a statistical fashion,
and hope to tease out clues to their origin and evolution as a population even if the
life stories of individual galaxies remain unclear.
Since H[ubble's publication of The Realm of the Nebulae in 1929, in which
the observational evidence was presented showing that these "nebulae" were indeed
galaxies like our own, the basic questions have centered around the determination of
* The epoch of galaxy formation;
* The processes which gave rised to the great variety of luminosities, morphologies, and constituent stellar populations of galaxies; and
* The relative effects of the initial conditions at the time of formation ("nature"),
and the environment of galaxies at later times ("nurture").
The study of the formation and evolution of galaxies (including that of our
own) is itself evolving very rapidly and recent, comprehensive reviews are scarce. For a
picture of the observational status of the field I suggest the contributions in Thronson
& Shull (1992) and the review paper of Koo & Kron (1992). For theoretical fare, the
lecture notes of White (1993) are highly recommended. While there is no substitute
for the detailed study of individual galaxies, particularly at high redshift, a timehonored approach has been to study galaxies in a statistical fashion as a population
or set of populations. This is motivated by the optimistic conviction that although
these objects may be the product of a random formation process, the statistics of
that process are well defined and contain important information about both galaxy
formation and cosmology. In this thesis, as well as in this branch of study, the galaxies
of interest are so-called 'normal' galaxies which do not exhibit signs of extraordinary
behavior such as an intense starbursts or the activity of a central nucleus or black
hole.
A first order description of galaxies is their luminosity and their number distribution with luminosity is called a "luminosity function" (LF). Luminosity is a
complicated convolution of the total mass of stars in the galaxy, the star formation
history, and the distribution of forming stars with mass (called the Initial Mass Function or IMF), and it means different things for different types of galaxies. The light
of early-type elliptical galaxies and the bulges of disk galaxies, in which significant
star formation is absent at the present epoch, is dominated by evolved stars who
have left the main sequence and give these objects their characteristic redder colors.
The luminosity of late-type disk or irregular galaxies arises in large degree from the
blue, short-lived 0, B, and A stars at the massive end of the main sequence, causing
these galaxies to be bluer and their luminosities highly dependent on the star formation rate. While an exact interpretation of luminosity is somewhat ambiguous, it
exhibits a strong correlation with the depth of a galaxy's gravitational potential well
and therefore its mass, as measured by the stellar velocity dispersion (in elliptical
galaxies) (Faber & Jackson 1976) or the circular velocities of neutral hydrogen (in
spiral galaxies) (Tully & Fisher 1977), and in both cases being closely proportional
to the fourth power of the velocity. Luminosity is also a relatively simple quantity to
measure.
To orient the reader in the succeeding discussions, I encapsulate the observational "facts of life" about galaxy luminosities and morphologies in the following
statements:
* Low-luminosity (dwarf) galaxies are more numerous than high-luminosity (giant) galaxies;
* Galaxies much more luminous than a certain cut-off value (- 1010L®) are exceedingly rare;
* Galaxies with different morphologies do not have the same LF;
* Relative representation among the different morphological types varies with the
local galaxy density: Elliptical galaxies are more numerous in very dense areas
and spirals more numerous in the mean field;
* Concentrations of gravitationally bound galaxies, called galaxy clusters, are the
unique home to Brightest Cluster Galaxies, a distinct class of ultraluminous
(_ 1011 -- 10 12 L®) giant elliptical galaxy, sometimes possessing an extended
halo.
As I shall show, measurements of the luminosities and morphologies of galaxies,
particularly in dense clusters of galaxies, can play a fundamental role in deciphering
the mechanisms responsible for these galaxy properties.
1.1
LUMINOSITIES & MORPHOLOGIES AS ARTIFACTS...
Any viable theory of galaxy formation and evolution must be able to reproduce the
essential characteristics of the galaxy LF, its dependence on galaxy morphology, and
its variation with environment. For the interested reader, some details of the theory
of galaxy formation and evolution as they relate to luminosities and morphologies, as
well as the relevant references, appear in Appendix A.
The standard theories of galaxy formation propose that the formation of structure in the Universe has proceeded hierarchically and has been dominated by a dissipationless or nearly-dissipationless "dark matter" component which forms gravitationally bound structures or "halos" on larger and larger scales with time. Galaxies
are formed when baryonic gas is able to fall into these halos and cool faster than
shock heating and pressure are able to bring it to hydrostatic equilibrium (White &
Rees 1978). This scenario can be tuned to predict a maximum mass (and luminosity)
to galaxies that is consistent with observations, but it also predicts that the galaxies
will form at the smallest size possible, in conflict with observations. To salvage this,
one theory incorporates feedback from star formation in the form of energy input from
the supernovae explosions of a first generation of massive stars (White & Rees 1978).
This still predicts a large number of faint galaxies relative to bright ones, such that
the slope of the LF is steep.
A second theory prevents the early formation of too many dwarf galaxies by
supposing a pre-heating and subsequent adiabatic cooling of the intergalactic medium
such that halos can only 'trap' gas when the equivalent temperatures of their gravitational potential wells are 'hotter' than the IGM (Blanchard et al. 1992). A flatter
LF is produced in this scenario. Thus measurements of the LF slope could be used
to rule out one or both of these theories.
The theory must also be consistent with the observed evolution (or lack of
evolution) of galaxies. The luminosity evolution of galaxies is divided into two components: The passive evolution of the original stellar population as it ages and active
evolution due to additional star formation, merging with other galaxies, or accretion
of satellite galaxies. Passive evolution will make galaxies appear brighter with increasing redshift, whereas merging or accretion will make them appear fainter. Star
formation taking place more than - 100 Myr (the main-sequence lifetimes of massive
stars) in the past will make galaxies appear brighter. Since it is impossible to identify
exact distant counterparts to the ancestral states of galaxies at the present epoch, the
evolution of the population as a whole must be examined statistically through the use
of a LF. Passive and active evolution of the galaxies will appear as a systematic shift
of the entire LF with redshift and a measurable brightening or fading in the cut-off
luminosity. Merging, which will deplete faint galaxies and supply brighter ones, can
also distort the shape of the LF.
The theory must also be able to provide a plausible explanation for the existence of the extraordinarily luminous Brightest Cluster Galaxies. These objects are
giant, elliptical galaxies, often with highly flattened outer isophotes that appear to
lie at or near the bottom of the gravitational potential well of the host galaxy cluster; Some have multiple nuclei and others have giant outer envelopes (Tonry 1987;
Schombert 1988). If BCGs formed by some special process involving the collapse of
the galaxy cluster core at a very early epoch then presumably their luminosity would
be independent of the state of the cluster at the observed epoch. If BCGs evolve
by cannibalizing giant galaxies which spiral to the cluster core under the influence of
dynamical friction (Ostriker & Hausman 1977; Hausman & Ostriker 1978), or by a
rain of debris from the collisions of galaxies higher in the cluster potential (Malumuth
& Richstone 1984), a correlation with global cluster properties would be expected,
i.e., one would expect to see less luminous prototypes in younger clusters . However,
the BCGs may asymptote to some final state where the core of the cluster has essentially been depleted of galaxies able to merge with the BCG and the monster, gorged,
evolves quiescently thereafter (Merritt 1985).
There are essentially two competing theories for the origin of morphological
types (the 'Hubble sequence'). The 'nature 'hypothesis holds that differences in the
initial density fluctuation field around galaxies led to variations in the angular momentum of the cooling gas. Low angular momentum gas went into the formation of
spheroidal systems while high angular momentum material formed disks. The 'nurture' hypothesis proposes that elliptical galaxies form from the collision and merger
of disk galaxies. These theories are not exclusive and may, in fact, both be true to
some degree. The contrast in morphological types is most pronounced between dense
clusters, where gas-poor early-type elliptical and SO galaxies predominate, and the
mean field, where the gas-rich late-type spiral and irregular galaxies are in the majority. This observation does not discriminate between these two theories since both
the initial conditions and the present-day environment of galaxy clusters is different.
However, one can differentiate between the two if the clusters could be 'age-dated'
by other observations. If environment plays a dominant role, youthful clusters would
have a higher fraction of late-type galaxies than older clusters.
1.2
... AND AS TOOLS
The luminosity function (LF) is not only a testable prediction of galaxy formation
and evolution theories, it is an important observational tool: The integrated LF is
the mean cosmic luminosity density PL, which, when multiplied by a mass-to-light
ratio of choice gives the fractional baryonic mass contribution to the critical closure
density;
2
QB-0.0027h- (M/L)® 1.9 x 108PL
LOMpc - 3
5
which, for example, can be compared to constraints from primordial nucleosynthesis.
The LF has played a central role in the debate over the interpretation of counts
of galaxies to faint magnitudes and the "faint blue galaxy problem".
Blue-band
number counts fainter than B-19 reveal a population of objects with an inferred comoving number density 3-5 times that of the galaxies of the present epoch (Peterson
et al. 1979; Tyson 1988; Jones et al. 1991). Paradoxically, the redshift distribution in
deep surveys is consistent with little or no evolution (Colless et al. 1990; Glazebrook
et al. 1995) and samples of high-redshift luminous galaxies show little evolution in
either counts or luminosity (Steidel et al. 1994). This has led to proposals that a high
co-moving number density of more rapidly evolving dwarf galaxies is responsible for
the excess in counts, a population that would manifest itself as a steep "faint end"
in the galaxy luminosity function (Koo et al. 1993; Treyer & Silk 1994; Gronwall &
Koo 1995). This possibility has in part motivated attempts to extend measurements
of the LF to very low luminosity levels.
At the other extreme, the observation that the intrinsic luminosities of the first-
ranked or Brightest Cluster Galaxies in clusters are extremely homogeneous, with a
dispersion of about 0.25 magnitudes (Humason et al. 1956) has led to their use as
astronomical 'standard candles' for the purpose of distance measurement (Postman
& Lauer 1995), tests of world models (Schneider et al. 1983a), and the measurement
of peculiar velocities (Lauer & Postman 1994). If somehow the homogeneity of BCG
luminosities could be understood, and the effects of merging discounted or accounted
for, observations of BCGs to high redshift could be a powerful test of combined models
of cosmology and stellar evolution. Along the same lines, evidence for the uniformity
and quiescent evolution of elliptical galaxies in clusters is now being considered for
similar tests (Pahre et al. 1996).
1.3
1.3.1
TECHNIQUES & PREVIOUS RESEARCH
LUMINOSITIES
In astronomy luminosity is usually expressed as an absolute magnitude M in the
wavelength pass-band of the observations
M = -2.5 log L + M®,
(1.2)
where L is the luminosity in solar units and Mo is the absolute magnitude of the Sun
in the observed range of wavelengths. For a 'red' (A - 7000A) magnitude, MR = +4.3.
The most systematic method of measuring the luminosity of galaxies is to measure
their flux (expressed as an apparent magnitude m) and their recessional velocity
cz, and calculate a rest-frame absolute magnitude using one's favorite description of
space-time. In a Robertson-Walker Universe,
M=m-Slog
qz+
h
qo+
42.39 - K
(1.3)
where h is the Hubble constant in units of 100 km sec - ' Mpc-', qo is the deceleration
parameter (specified by the cosmology, i.e. qo = Q/2-A), and K is the "K-correction"
which accounts for the narrowing of the pass-band in the galaxy rest-frame and restshifting of the source spectrum with respect to the observed band; it is almost always
a positive quantity.
Upon acquiring the absolute magnitudes of the desired number of galaxies in
a survey over a given region of the sky, a LF, q(M), is constructed by weighting each
galaxy by the volume of space in which it could have been detected: If the ith galaxy
in the absolute magnitude interval M -+ M + dM could have been detected to a
redshift zi over a solid angle Qi(z)
2-1
3
O(M)dM = (3000Mpc)-3 x h'E
(Joi
dzQ,(z)
+
o +1)(1z)
q04(1 + z) 1
+ 2qoz
1)1+
(1.4)
Many different methods are actually used to construct luminosity functions but this
expression conveys the essential mathematics. This and similar operations can be
performed on redshift surveys to find the local LF (Sandage et al. 1979; Efstathiou
et al. 1988), or, if one is more ambitious, on deeper surveys to find the evolution of
LF with redshift (Eales 1993).
Beyond the monumental task of acquiring magnitudes and redshifts of a sufficient number of galaxies to adequately sample the luminosity function, there are
potential pitfalls in the standard technique of LF construction. These arise from a
combination of the apparent magnitude limit of the surveys and their unrestricted
depth. Two of the most important can be elucidated by a careful inspection of Eqn.
1.4. In magnitude-limited surveys the most luminous galaxies can be seen to much
larger distances than the least. An interesting range of galaxy luminosities over which
to construct O(M) is - 5 magnitudes, a factor of 100, for which the corresponding
ratio of Euclidean volumes is 103.
The sampling of low-luminosity galaxies, which
can be detected only in a much smaller regions of space, suffers from larger Poisson
noise. Furthermore, the sampling becomes sensitive to variations in the galaxy density (large-scale structure).
For example, if an observer inhabiting a region of low
mean density were to construct a LF, this intrepid astronomer would mistakenly find
disproportionally fewer dwarf galaxies compared to giant galaxies, which are sampled
from a larger, more "fair" volume.
Secondly, galaxies are seen against a noisy background due to the brightness
of the night sky and the intrinsic noise of the detector. They are thus detected and
selected not just on the basis of a total integrated flux, but according to surface
brightness as well. Low surface brightness (LSB) galaxies are more difficult to detect
and even more difficult to obtain accurate magnitudes and redshifts for. The effect is
redshift-dependent due to cosmological dimming and K-corrections, and thus the same
galaxy population may not be observed over the entire depth of a survey. Galaxies
with anomalously low surface brightnesses but respectable luminosities have been
discovered (McGaugh et al. 1995) but their space densities appear to be small relative
to their brighter counterparts and their importance is debatable. Still, one can still
argue that volume and surface brightness biases make for significant uncertainties in
present field galaxy LFs (Driver & Phillips 1996).
There are other, as yet unquantified systematics as well. For example, positive
luminosity evolution combined with the volume bias will cause the most luminous
galaxies in a magnitude-limited survey to be too bright, effectively "stretching" the
LF. Galaxies of different rest-frame spectra will suffer different K-corrections, which
can induce significant errors if the spectra, especially the ultraviolet spectra which
are red-shifted into the observed rest-frame, are not well known enough to make
an accurate correction. All these systematics become more serious with increasing
redshift.
A complentary scheme to construct the LF, pioneered by Abell, e.g., (Mottmann
& Abell 1977), is to exploit the highly non-uniform distribution of galaxies in space
and survey the densest concentrations of galaxies, called galaxy "clusters". These
clusters are typically a factor of
103 denser than the cosmological mean over a
volume of a few Mpc3 . They create a significant projected surface density of galaxies
on the sky despite intervening distances of hundreds of Mpc. The ratio of the size
of clusters (- 10 Mpc) to their distances (- 500 Mpc) is small, meaning that errors
in luminosity of at most a few percent occur by assuming the galaxies are all at the
same distance. The high projected surface density also permits efficient imaging and
photometry of a large number of galaxies in a limited field of view. Moreover, the
galaxies are all observed at the same rest-frame surface brightness limit modulus their
different K-corrections. If the redshift of the cluster is known, galaxy luminosities
can then be calculated using Eqn. 1.3 in a kind of "poor man's redshift survey".
Of course, without spectroscopy to confirm the redshifts of individual galaxies,
it is impossible to be certain whether any particular galaxy is a member of the cluster
or a field galaxy projected along the line of sight. However, it is possible to measure
the statistical distribution of galaxies with some measured property (such as luminosity) by subtracting the mean contribution of background galaxies. The brightest
galaxies associated with the nearer clusters are nearly all cluster members because of
the relatively low numbers of field galaxies at the same apparent magnitude. However, the background contribution is significant and must be carefully estimated in
the case of more distant clusters and their lower-luminosity members.
It is also important to point out that, without an independent measure of the
total projected overdensity along the cluster line-of-sight, it is not possible to correctly
compute the normalization of the cluster LF and compare it to the field; Only the
shape of the LF can be accurately determined. Although a complete treatment of
the LF problem must include issues of normalization, the observational part of this
thesis is concerned with a construction of the cluster LF and I do not deal with this
issue further.
Parametric descriptions of the galaxy LF attempt to capture the basic elements of the distribution using simple functions and are useful in comparing between
individual surveys, and between surveys and models. Abell (1962) used a broken
power law to describe the abundance of faint galaxies and the limits on bright ones.
Schechter (1976), inspired by the description of the mass distribution derived in Press
& Schechter (1974), introduced an analytical description of the LF now commonly
called the Schechter function;
n(L)dL , e- L / L *
n(L~i-
~LI.
dL
(L*jL\
(1.5)
where L* describes a characteristic cut-off luminosity (or absolute magnitude) and a
the power-law distribution of low-luminosity galaxies faint galaxies.
The Field: Hubble (Hubble 1936) constructed the first galaxy luminosity
function (a Gaussian) from a small, and very biased, sample of nearby galaxies.
Much more recent surveys (Kirshner et al. 1979; Tully 1988) and their re-analysis
(Efstathiou et al. 1988) have found Schechter-like LFs with flat faint-end slopes (aN 1). Most notable are the recent very wide field surveys which have found similar
results. Using the Mt.
Stromlo-Automatic Plate Machine (APM) survey of field
galaxies selected from blue photographic Schmidt plates, (Loveday et al. 1992) found
a, flat distribution with a = -0.97 ± 0.10 to MB < -19.
(Marzke et al. 1994) also
obtained a = --1.0 for Center for Astrophysics (CfA) redshift survey galaxies over the
range -20 < Ms < -18 but found evidence for an upturn in the LF at -16 + 5 log h.
The reality of the latter result is difficult to ascertain due to uncertainties in the
systematics of the Zwicky magnitude system used. The combined samples in the
AutoFib survey (Ellis et al. 1995) give a z < 0.1 LF with a = -1.09 ± 0.10. An
evolution of a towards steeper values at higher redshift is also claimed. Analyses of
the recently completed Las Campanas Redshift Survey (LCRS) include luminosity
functions constructed from subsets of the 23690 galaxies in the survey at z - 0.1
(Lin et al. 1996). Due to a 'dip' in the LF, this survey finds a formal best-fit slope
of a = -0.70 ± 0.05 to Mr < -17.5,
but the data is better described over-all by
a- - 1.0, and a possible upturn at Mr > -17.
Clusters: Schechter (1976), fitting his empirical form to the photographic
data of (Oemler 1974), found a = -1.25 (with an error greater than 0.2) and MA =
*-19.1 ± 0.2. The relatively small sample sizes of typical cluster samples and the
covariance between M* and a led subsequent photographic research on the cluster
LF to adopt the Schechter value for a in order to obtain a well-defined value of
.M* (Dressler 1978b; Lugger 1986; Oegerle & Hoessel 1989). Derivations of the LF
from photographic data of the Virgo and Fornax (Ferguson & Sandage 1988) give
a steeper slope; a-
- 1.3 and ML- - 21. More recent work using CCD imaging
(Driver et al. 1994; Driver et al. 1994; Kashikawa et al. 1995; De Propris et al. 1995)
has found a steep faint-end or an upturn in the slope with values of a approaching
-1.4 to -2. On the other hand, deep CCD imaging in nearby clusters finds either an
upturn at faint absolute magnitudes well below the limits of field surveys (Bernstein
et al. 1995), or a simple flat LF (Chiboucas et al. 1995).
To summarize, there is agreement among all the low-redshift surveys of field
galaxies that the LF is flat (a = -1) to an absolute magnitude of about -17.
Fainter
than that there may be an upturn or excess of dwarfs. In contrast some work on
clusters finds a steep LF at much brighter magnitudes (Driver & Phillips 1996). In
Fig. 1 I plot various Schechter function parameter pairs obtained for surveys of the
field and clusters. This plot is intended to sketch out the playing field, and is not
intended to be complete nor unbiased. All absolute magnitudes are tranformed to
Gunn r magnitudes using the assumptions B - r = 1.1 for field galaxies, B - r = 1.5
for cluster galaxies, F - r = -0.58 for cluster galaxies, and R - r = -0.09 for
cluster galaxies. The dashed line approximately describes the covariance between M*
and a such that points on loci parallel to this line probably describe very similar
distributions.
As mentioned earlier, redshift and photometric surveys suggest little or no
luminosity evolution of normal galaxies in the field to z - 1. Studies have also focused
on cluster galaxies where the case for or against luminosity evolution, particularly
merging at the bright end of the LF and the brightest cluster galaxy, is not so clear.
The results have typically been cast in the form of the difference between M
1 , the
absolute magnitude of the first-ranked galaxy or BCG, and some fainter magnitude
such as M2 or M*. Merging would make more luminous galaxies grow at the expense of
less luminous ones, widening the gap between them. The effect is expected to be most
pronounced at the bright end since dynamical friction, the process driving merging, is
proportional to the mass of the system. Some researchers found evidence for an effect
(Dressler 1978a) while others (Gunn & Tinsley 1976; Schombert 1984; Lugger 1986;
Oegerle & Hoessel 1989) could not confirm this. A significant correlation with the
morphology of the cluster (Rood-Sastry and Bautz-Morgan type) such that clusters
with more relaxed morphologies which are supposed to be dynamically 'older'have a
brighter M1 and fainter M*, is now claimed by Trevese (1996). A recent CCD survey
of 119 BCGs in z < 0.05 Abell clusters (Postman & Lauer 1995) finds no correlation
with Abell richness (however, their sample only included R < 3 clusters).
1.3.2
MORPHOLOGIES
Although the morphologies of nearby galaxies have been differentiated in botanical
detail, work on the the morphologies of large samples of more distant galaxies in both
the field and clusters has lagged behind construction of luminosity functions. Until recently much of the structural classification was done by eye from photographic
plates, (Dressler 1980; Schechter & Dressler 1987). Such studies made clear that elliptical and SO galaxies (disks with old stellar populations) dominated the population of
clusters. A tight correlation between local density and galaxy morphology was found
(Dressler 1980), although the exact interpretation of this finding has been disputed
(Whitmore et al. 1993). Information on more distant field and cluster samples has
been restricted to colors, with blue galaxies being equated with disk galaxies and red
galaxies with ellipticals. This data suggests interesting evolution in the galaxy population with redshift, including the excess of faint blue field galaxies discussed above.
In galaxy clusters, the fraction of blue galaxies was found to rise dramatically above
a redshift of - 0.2 (Butcher & Oemler 1984). Spectroscopic investigation (Dressler &
Gunn 1983; Dressler et al. 1985) and later high-resolution imaging by HST (Dressler
et al. 1994) suggest that a large fraction of these are disk galaxies undergoing an increased episode of star formation. It is not yet clear whether this process is somehow
connected to the relative abundance of ellipticals in clusters at the present epoch.
Steps are being taken to rectify the situtation and develop schemes that
can be applied to digital surface photometry, including bulge-disk deconvolution
of nearby spiral galaxies (Schombert & Bothun 1987), neural-network classification
(Odewahn 1995), and principal component analysis of the surface-brightness profiles
(Watanabe et al. 1985; Han 1995). For more distant objects the schemes are split
among using simple characteristics such as surface brightness and central concentration of the flux (Okamura et al. 1984; Doi et al. 1993; Abraham et al. 1994; Doi
et al. 1995) and, for higher-resolution data, fitting surface brightness profiles (Casertano et al. 1995). This field of distant galaxy classification is in its youth and has only
begun to explore the possibilities opened up by Hubble Space Telescope data and large
digital surveys, as well as the combination of spectral and structural information.
1.4
MOTIVATION FOR THIS THESIS
It seems clear that there is much work to be done before a significantly better understanding of galaxy formation and evolution is in hand. An interesting contribution
can be made by a more systematic survey of cluster galaxies using the wide-field CCDs
now available to image tens of thousands of galaxies to high photometric precision, a
primary goal being to construct a high signal-to-noise LF and measure the slope of
the faint end.
Furthermore, if one can somehow "age date" these clusters using, for example,
an estimate of their over-all morphology independent of the properties of the individual galaxies, a number of interesting evolutionary tests can be made, including
changes in the faint-end slope, the absolute magnitude of the cut-of M*, the relative
numbers of disk galaxies, and the luminosity of BCGs. This, in true Don Quixote
style, is what I have set out to do in this Thesis.
-1.4
-1.2
-1
-0.8
--n P
• 20
-20.5
M
-21
-21.5
- 5 log h
Fig. 1.- Field and Cluster LF - Schechter parameters
Schechter function parameters for field (open points) and cluster (solid points)
surveys. The dashed line indicates the direction of covariance. (a) HI spirals
(Tully 1988); (b-c) Fornax & Virgo (Ferguson & Sandage 1988); (d)KOS (Kirshner
et al. 1979); (e) 3C295 (Mathieu & Spinrad 1981); (f) combined surveys (Efstathiou
et al. 1988); (g) Abell Cl (Trevese et al. 1996); (h) Abell Cls (Dressler 1978a); Abell
Cls (Lugger 1986); (i) Abell Cls (Oegerle & Hoessel 1989); (j) Abell Cls
(Schechter 1976); (k) Autofib (Ellis et al. 1995); (1) APM/Stromlo (Loveday
et al. 1992); (m); DARS (Peterson et al. 1986) (n) LCRS (Lin et al. 1996); (o)
SSRS2 (Da Costa et al. 1994); (p) CfA (Marzke et al. 1994); (q) A576 (Mohr
et al. 1996);
Chapter 2
A Wide-Field Optical Survey of Cluster
Galaxies
An optical imaging survey of rich clusters of galaxies was conducted to efficiently study
the distribution of a large number of member galaxies with luminosity and morphology
and establish any correlation between these properties and that of the host clusters.
The large number of clusters (67) in the survey and the relatively wide field surveyed
in each cluster makes this a unique resource with which to undertake this research: I
estimate that there are of order 105 cluster galaxies in the survey. In §2.1 I describe
the survey and the construction of the galaxy photometry catalog. The measurement
of the background field galaxy counts, crucial to a statistical determination of the
galaxy luminosity function, is presented in §2.2. The global properties of the galaxy
clusters, i.e., density and morphology, are discussed in §2.3. The galaxy LF for these
clusters is derived and its variation with redshift and cluster properties explored in
§2.4; The brightest cluster galaxies analyzed separately in §2.5. In §2.6 I introduce a
new technique to characterize the surface-brightness morphologies of distant galaxies
and analyze the relative distribution of disk and spheroidal systems in the clusters.
2.1
DESCRIPTION OF THE SURVEY AND GALAXY CATALOG
2.1.1
CLUSTER SELECTION
The clusters were selected from the ACO catalog (Abell et al. 1989) containing 4076
such objects in the Northern and Southern hemispheres and essentially an extension of the original Abell catalog (Abell 1958). The Abell catalog was constructed
from visually selected enhancements in the projected galaxy distribution on Palomar Observatory Sky Survey (POSS) plates of the Northern Hemisphere obtained
at the Caltech 1.2 meter Schmidt telescope, and Southern Sky Survey plates taken
at a similar telescope in Australia. By definition, each cluster contains at least 30
galaxies with apparent magnitudes in the range m 3 and m 3 + 2 (where m 3 is the
magnitude of the third-ranked galaxy) in an aperture of radius 1.5 h -
1
Mpc of the
cluster center. A more complete sub-catalog of "rich" clusters with at least 50 members was also defined. Altogether, Abell defined six richness classes (0-5) in order
of increasing membership. Only a small fraction of the ACO cluster have published
redshifts (Struble & Rood 1991). Nearly all Abell clusters with measured redshifts
lie at z < 0.3.
As is the case with any sample of astronomical objects selected by the human
eye, the statistics and limitations of the catalog are not well understood. Without
distance information, there is no one-to-one correspondence between the galaxy distribution in space and the observable; the projected surface density on the sky. Such
a catalog could be contaminated by spurious sources caused by the large-scale structure of the galaxy distribution. A gravitationally bound mass of galaxies does not
necessarily have to appear as a region of high surface density on the sky, and vice
versa. For example, a number of less massive but more common enhancements in the
space density could be (un)fortuitously aligned along a line of sight, mimicing the
existence of a much more massive object. By the same token, an underdense region
along the line of a genuine galaxy cluster could suppress the cluster signal and pre-
vent its detection. Most Abell clusters have no spectroscopic confirmation of being
a significant overdensity in the space distribution of galaxies, nor even a measured
redshift.
However, studies of the distribution of ACO clusters with richness find a sudden decrease in the their number below a galaxy count of 50, i.e., "poor" R - 0
clusters (Ebeling 1993). This suggests that the catalog is seriously incomplete for
these objects. Clusters can also be identified by the X-ray emission from a hot intracluster medium in hydrostatic equilibrium with the gravitational potential (see Ch.
III). Extensive studies of the Abell clusters detected in the All Sky Survey of the
Rosat X-ray observatory satellite find that nearly all of the richer (R > 0) Abell clusters are physical objects and not the result of projection effects (Ebeling et al. 1993;
Ebeling 1993) (see Ch III). Thus, with a few exceptions, I have included only R > 0
clusters with measured redshifts in my sample.
Clusters were selected with redshift to optimize the efficiency of galaxy photometry to a fixed rest-frame surface brightness.
As discussed in §1.3, surface-
brightness selection effects may be an important culprit in biasing number counts
and redshift surveys. All other things (such as richness) being equal, distant clusters
will subtend a smaller angle on the sky and thus more of their galaxies will be included in the fixed field of view of an imaging camera. However these same galaxies
will have a lower observed surface brightness due to cosmological dimming and the
red-shifting of the rest-frame spectra (K-correction).
I gauge the efficiency of observing cluster galaxies at a certain redshift by
estimating the specific integration time required per galaxy to achieve a fixed restframe surface brightness limit. I assume that the density of galaxies around the cluster
center decreases as the radius squared, such that their projected surface density at
angle 0 from the cluster center falls as 01 (Beers & Tonry 1986).
In arc-second
seeing, cluster galaxies will be resolved to z - 0.3, the limit of the Abell catalog and
thus observing to a fixed surface brightness is sufficient to observe the same galaxy
population, independent of redshift. For a fixed cluster richness, the total number of
cluster galaxies will be proportional to the field of view. The cosmological surface
brightness dimming is a factor (1 + z)- 4 . I adopt a simple factor of (1 + z)- 1 for
the K-correction of elliptical galaixes (Postman & Lauer 1995),. The K-corrections
of ellipticals are the largest in the chosen observing band-pass (R) and redshift range
of the Abell catalog; Correcting for elliptical galaxies will result in generous overcorrections for all other galaxy types. Galaxy photometry is limited by the Poisson
noise of the night sky background and the signal-to-noise of a measurement increases
with the square-root of the integration time. The specific integration time per galaxy
is then
tint -
(1 + z)
10
z-
2
(2.1)
,
whose inverse is plotted in Fig. 1. There is a maximum at z = 0.25 and a broad
range of suitable redshifts from z > 0.1. I tended to select clusters close to z = 0.1 to
maximize the amount of spatial information availabel for the classification of galaxies
(§2.6).
The sample was limited to clusters with Galactic latitudes
Ibl
> 300. No other
selection criteria such as morphology were used. The selection is essentially unbiased in that I did not intentionally select clusters that have already been extensively
observed. The drawback is that there is the possiblity of non-existent clusters, erroneous positions or incorrect redshifts in my sample. I obtained Charge-Coupled
Device (CCD) imaging through red (5500-8000 A) filters. This wavelength range was
chosen because galaxies, particularly the elliptical galaxies that dominate the cores
of clusters, emit most of their light at redder wavelengths, and the effects of redshift
on galaxy spectra are small. Exposure times were generally selected to achieve a
constant surface brightness limit although this is not always the case. Most of the
imaging covered a 8.4 arc-minute field of view which, at z = 0.13, corresponds to 1
/h- 1 Mpc. Some clusters were imaged in a mosaic pattern to obtain a 16 arc-minute
field of view.
2.1.2
OBSERVATIONS
The observations were made at the Michigan-Dartmouth-MIT Observatory using the
2.4 meter Hiltner telescope with a f/7.5 secondary. The four observing runs took
place in May, 1994; September, 1994; May, 1995; and October, 1995. The first five
days of the first run used a camera with a Loral 10242 CCD that is thinned and
back-side illuminated in conjunction with a special Rs filter which has a redder and
wider band-pass than a standard KPNO R interference filter. The plate scale was
0.275 arc-seconds per pixel giving a field of view of 4.6 arc-minutes. All subsequent
observations used a camera with a STIS (Space Telescope Imaging Spectrograph)
thick, front-side illuminated CCD with a 20482 format, 0.24 arc-second pixel scale,
and a 8.2 arc-minute field of view. Due to the unavailabity of a larger Rs filter, an
AURA R filter (similar to KPNO R) was used. Plots of the R and R, bandpasses
are shown in Fig. 1.
The weather during 6 of the 8 nights of the May 94 run was judged to be
photometric based on repeatability of the measurements of standard stars and visual
inspection of the sky for clouds. Twenty-two clusters were imaged under photometric
conditions and they serve as the primary data for this thesis. No photometric nights
occured for subsequent runs and no usable data was obtained at all during the last
run (October 1995). Usable data were obtained for 67 Abell clusters. Table 1 lists
the Abell clusters surveyed, giving the telescope and instrument, total field of view,
and exposure time. It is also indicated whether the imaging was obtained under
photometric conditions.
2.1.3
IMAGE PROCESSING
Raw CCD images require extensive processing to remove systematic effects and artifacts to obtain a "clean" image suitable for galaxy photometry analysis. I used
the standard routines in the Image Reduction and Analysis Facility (IRAF) for this
purpose.
The signal from a CCD camera is biased by an arbitrary amount sufficient to
ensure that the negative-going fluctuations from noise are not truncated. This bias
level is recovered by reading each CCD column beyond the imaging area and creating
an "overscan" strip. The bias level along the overscan is fit by a polynomial and
subtracted from the image; the overscan region is then trimmed off.
Flat-fielding removes variations in the photometric zero-point across a CCD
image. These variations arise from vignetting, obscurations by objects and dust
particles in the optical path, and CCD pixel-to-pixel response variations. Images
of blank fields such as screens on the inside of a telescope dome or of the twilight
sky are sometimes used as flat fields to correct CCD images. However, these do
not always faithfully reproduce response differences because of the effects of lighting,
contaminating clouds or stars, and poor reproduction of the color of the night sky.
More accurate flat fields were constructed by averaging over 20-40 images of different
fields, excluding the highest few pixel values in the averaging process at each pixel.
This technique relies on most of the fields being occupied by dark sky such that
elimination of the few highest pixel values eliminates any contaminating objects.
Charged particles from cosmic ray and the decay of trace radioactive materials in the instrument produce spots in the CCD images with a characteristic highly
peaked signal and a small footprint. In thinned CCDs the CPEs are typically elongated trails while in thick CCD's they are circular points. The CPEs accumulate
on the image at a constant rate while an exposure is being made and ultimately
limiting practical exposure times. CPEs coinciding with galaxy images can cause
substantial photometry errors. If they are sufficiently numerous they may also make
automatic determination of the image point-spread function difficult. CPE elimination is a critical step for facilitating the automatic analysis of the survey images.
CPEs can be eliminated from very deep images by breaking the exposure into many
multiple, shorter exposures. The events are uncorrelated from image to image and
can be removed by median filtering. the exposure times used for this survey (15-40
:minutes) and the time required to read out the CCD make this option impractical.
Here, CPEs are removed using the COSMICRAYS routine in the NOAO package
installed on IRAF. The galaxy detection and classification routines (discussed below)
are also able to descriminate against CPE events.
The COSMICRAYS routine examines each pixel which exceeds a certain threshhold above the background and compares its value to the average pixel value in a 7 x
7 box centered on the pixel in question. If the former exceeds the latter by a certain
factor then the pixel is identified as a cosmic ray and its value replaced by the mean
of four neighboring pixels (excluding the second highest pixel). The process can be
repeated to iteratively clean all CPE-effected pixels. A smaller box reduces the effect of gradients in the image, while a larger box is required for CPEs with multiple
pixels. Less stringent criteria (a smaller factor) will be more efficient at removing
CPEs but may also introduce errors in the photometry, i.e., central pixels in objects
may be altered. Areas of the image with steep pixel value gradients are susceptible
to this effect. These will occur for point-like objects; extended objects like galaxies
will always have broader profiles. The broader the point spread function with respect
to the COSMICRAYS box, the lower the threshhold may be.
The minimum gradient threshhold for an image can be estimated by constructing a model PSF and evaluating the ratios of pixel value to averaged neighbor value.
I modeled the PSF as a Moffatt function;
I(r) =
1
F
2r M]
2/2-1
(2.2)
where FWHM is the full width half maximum of the PSF and / is a parameter which
governs the power-law behavior of the 'wings' of the PSF. Typically, /3
2.5. A
plot of the minimum flux ratio factor for CPEs versus the FWHM in pixels is shown
in Fig. 2. A typical FWHM at the plate scale of the survey is 4 pixels. Based on
these results I set the detection threshhold to 5o and the flux ratio to 10%. Sixteen
iterations were used to ensure that all pixels contaminated by CPEs were corrected.
"Bad" columns due to faulty CCD registers are removed by linear interpolation
across the columns. The final step is to align and sum any multiple images of a field
using the GEOMAP and GEOTRAN routines in IRAF.
2.1.4
PHOTOMETRY
Most of the observations were performed under non-photometric conditions, e.g.,
haze, thin cirrus or scattered clouds. In fact, reliably photometric conditions occured
only for 6 of the 8 nights on the first run and never on any of the three successive runs.
Nevertheless, this limited photometric data is critical for obtaining absolute magnitudes for M*, M1 and comparing between individual clusters and between clusters
and the field galaxy photometry of other surveys.
Photometric images were calibrated using observations of standard stars (Christian et al. 1985). These stars are in (or in the line of sight of) globular or open clusters,
which afford a sufficient number of stars for the relatively small field of view of CCD
cameras. Photometric observations with each of the filter/CCD combinations were
analyzed independently.
Stars are detected in the images using the DAOFIND algorithm in the the
DAOPHOT package of stellar photometry tools in IRAF. This routine convolves
the image with a Gaussian with 1 arc-second FWHM and finds peaks above a set
threshhold. Photometry is performed on each star by counting the total counts within
a 3 arc-second radius circular aperture and subtracting a fixed sky value determined
by using the 1[TERSTAT routine (described below). The use of a global sky value
for each image avoids contamination problems from adjacent stars. The 3 arc-second
radius was found by trial-and-error to provide a balance between including as much
of the scattered light as possible while minimizing the probability of contamination
by neighboring stars.
A convenient expression of the calibration is the magnitude mi of a star that
generates 1 DN/sec in the detector placed above the atmosphere. For the Loral
CCD + R, filter on the 2.4 meter telescope I found mi = 28.95 ± 0.05 for a star
with B - R =: 1.0 color. There is a small but measurable color term such that mi
increases as 0.1(B - R). The dissappointingly large error in the calibration possibly
arises from a signficant systematic scatter in the calibration point, to which the
problem of performing simple aperture photometry in the relatively crowded fields
near star clusters is probably a major contributor. I observed no measurable drift in
the calibration over 5 days of of photometric observations. The calibration for the
AURA R/STIS combination was found to be mi = 23.00 + 0.05 with no apparent
color term. Note that the - 1 magnitude difference in response between the thinned
and thick CCDs was compensated for by roughly doubling the exposure times for the
latter.
The zero-point magnitude for each individual image is found using the equation:
mo = mi + C(B - R) - aRz - gR
(2.3)
where the last two terms are the atmospheric extinction through an airmass of z and
the galactic extinction. Since no color information is available, I adopt a value of
B - R = 1.5, typical of cluster ellipticals.
An extinction coefficient of 0.10 was used, intermediate between the standard KPNO R values(-
0.12) and that for the redder Thuan-Gunn r filter of 0.08
(Kent 1985). Uncertainties of up to 50% in this value are relatively unimportant since
all of the photometric images were obtained through airmasses less than 1.3. While
errors in aR may produce small offsets in the absolute photometry, the errors in the
relative photometry between cluster images will be less than 0.01 magnitudes. Galactic extinction is calculated using the B-band extinction values given in the NASA
Extragalactic Database (NED) and multiplying these by 65% to account for the inverse dependence of extinction on optical wavelength (Lang 1980). If the value was
not available from NED I adopted a value based on inspection of reddening maps
(Burstein & Heiles 1982; Burstein & Heiles 1984).
Galaxy photometry is characterized by the existence of nearly as many photometry systems as there are surveys. To compare with surveys in these other systems it is necessary to find transformations between them. At red wavelengths, the
Kron-Cousins R and Thuan-Gunn r pass bands are used with CCD imaging and the
corresponding plate system is F magnitudes, usually converted to equivalent rF magnitudes. The magnitudes here are in the Kron-Cousins R system and to compare with
other work corrections for the different band-passes used must be made. Fortunately,
the corrections are not large; of order 10%.
Faint field galaxy counts are dominated by spirals at z - 0.2 - 0.5 while
the Abell clusters are more elliptical rich and lie at about half the distance. This
combination of different intrinsic spectra and different redshifts means that separate
corrections should be made for the field and cluster samples (I do not attempt to
tailor the corrections to the individual galaxy types within the clusters).
For field galaxies, I adopt R, - R = 0.14 (Steidel & Hamilton 1993) and
R - rF = 0.13 (Metcalfe et al. 1991). The zero point of R - r is estimated to be less
than 0.1 magnitudes with a color term of less than 0.2 magnitudes per B - R color
for LCRS field galaxies (Lin et al. 1996). On the other hand, CCD stellar photometry
(Jorgensen 1994) gives the transformation R = r - 0.11 (g - r) - 0.32 + 0.02, which for
a typical spiral galaxy with g - r - 0.3, gives R - r- - 0.35. I assume a zero point
similar to that of Lin et. al. (1996), but adopt the color term of (Jorgensen 1994) to
correct for the redder cluster galaxies, to get R - r = -0.09.
Absolute magnitudes are calculated from apparent magnitudes assuming qo =
0.1 and a K-correction of the form 2.5 logo1 (1 + z). The latter is used both by Lin et.
al. (1996) for field galaxies in the LCRS survey and by Postman & Lauer (1995) for
brightest cluster (elliptical) galaxies at z < 0.5.
2.1.5
IMAGE REDUCTION
Catalogs of galaxies and basic measurements of their distribution of surface brightness
were constructed from the images using the FOCAS routines developed by Valdez
(NOAO, 1984). These routines consist of the detection and construction of objects,
splitting objects with merged isophotes, separating stars and galaxies, and computing
magnitudes and surface brightness properties. The brightest cluster galaxy (BCG), or
galaxies if there were two giant ellipticals of roughly equal luminosity, were removed
by isophote fitting and model subtraction. These galaxies typically have extended,
low surface-brightness halos which are difficult for FOCAS to measure and which can
cause it to produce spurious multiple objects. Treatment of the BCG's is discussed
in §2.5.
The FOCAS DETECT routine convolves an image with a detection filter to
find locations on the image where there is a significant signal level, i.e., the weighted
sum of the pixel values above background exceeds the noise by a statistically significant amount. This level, set by the user but limited by the noise, is the detection
isophote level. The choice of detection filter has a significant impact on the signal
threshold at which different objects can be detected. A typical choice is a Gaussian
with a width approximating that of the image point-spread function. The detection
filter-convolved image is scanned line by line. Along each line, object "run length
codes" are created by finding contiguous sections where all pixels have values above
the detection threshhold. Objects are then assembled from overlapping run length
codes, where diagonally adjacent pixels are considered to be overlapping. The detection threshhold, scaled to the RMS noise of the sky background, is thus a compromise
between avoiding finding too many spurious objects and including more signal from
each object at lower signal levels. The FOCAS routines do not, unfortunately, include
special techniques to extend galaxy photometry to fainter isophote levels.
An optimal detection scheme is efficient at detecting the extended images of
galaxies, is not prone to finding spurious objects from the noise background, and
does not suffer from biases with image scale. Detection of spatially resolved galaxies
is substantially different than detection of point-like objects (stars). There are three
points in the FOCAS image reduction package where tuning to achieve optimal galaxy
detection can be accomplished. The first point is the design of the spatial filter used to
detect objects. This consists of a set of pixel weighting values wi which are convolved
with the pixel values at the location in the image of a prospective object in the field.
If the values of a collection of pixels are si and the uniform background noise has
RMS o, the total signal-to-noise will be
S/N =
wS
(2.4)
Differentiation leads to a condition on the filter weights w, which extremizes (maxi-
mizes) signal-to-noise;
Sk iW - Wk iWiSi = 0,
(2.5)
which is always satisfied if wi = si. In other words, the optimal filter to detect
a galaxy has the same profile as that galaxy. The drawback from using extended
detection filters is increased computational overhead and increased susceptibility to
systematic errors in the background (e.g., due to problems with flattening). And,
of course, one does not know a priori the surface brightness profiles of the cluster
galaxies.
Galaxies exhibit a wide range of surface-brightness profiles which have been
described as exponentials, power-laws, or more complicated functions (Mihalas &
Binney 1981). For the sake of simplicity, a reasonable choice is an exponential filter
with a scale length similar to those of disk-like galaxies. This will significantly improve
the detection characteristics over a Gaussian filter with the PSF FWHM and help
ensure that the survey is not biased against these galaxies with their lower central
surface brightness. A canonical disk scale length is 4 kpc; assuming Ho = 67 km
sec-lpc - 1 this is an angular scale length of 1.44 arc-seconds (roughly 5.5 pixels) at
z = 0.13. The angular width of the detection filter is adjusted according to the known
redshift of each cluster to avoid distance biases. Beyond a redshift of z = 0.25 the
FWHM of the filter is less than 1", i.e. smaller than the typical seeing disk, and
there is no advantage to decreasing the width. For higher redshift clusters the total
width of the detection filter template is set equal to the the "half-light" diameter of
the exponential profile, 3.5 scale-lengths.
The second optimization point is setting the minimum number of pixels above
the threshhold required to define an object. In theory, one can maintain rejection of
spurious objects at lower detection threshholds by increasing the minimum number of
pixels required. The trade-off is that it becomes increasingly likely that sources near
the detection limit will fail the minimum pixels test and be rejected. Real objects
with the smallest footprints will be point-like and thus I scale the required number of
pixels with the size of the seeing disk. I set the minimum number of pixels equal to
the area within the half-light radius of the PSF. For a Gaussian PSF, this translates
to
Amin
1.1W 2 ,
(2.6)
where W is the FWHM of the PSF in pixels. Amin is typically - 20 pixels for the
image scale and seeing conditions of the survey.
Finally, one can adjust the pixel value detection threshhold in terms of the
RMS noise in the image. As one lowers this threshhold one achieves detection of
lower surface brightness objects and more accurate isophotal magnitudes, but at the
cost of a higher rate of spurious detections. Extensive tests of the DETECT routine
with Monte Carlo simulations of noise images using different detection threshhold
levels and minimum areas show that the spurious detection rate per pixel obeys the
empirical relation;
N-9
x 104eU-
2
(A/12)o° 1 8 ,
(2.7)
where a is the detection threshhold and A is the minimum area in pixels. Thus A must
increase very rapidly with a to maintain the same number of spurious detections. An
acceptable rate (1% spurious/real) is attained using the suggested threshhold of 2.5a.
Note that these spurious objects will almost all lie at or near the limiting magnitude.
Although FOCAS provides a SKY routine to find the local background level
around each detection limit, the cluster images are very well flattened and a single
global value for the background is sufficient. This also avoid problems with object
contamination of the background apertures in crowded fields. A background level
for each image is calculated in IRAF using the ITERSTAT routine developed by M.
Dickinson and kindly provided by C. Steidel. This algorithm calculates sky values by
iteratively calculating mean and standard deviation and rejecting high-valued pixels
associated with objects.
Deep images of clusters are crowded and galaxy isophotes may overlap. Overlapping galaxies will be detected as a single object by the DETECT algorithm. The
SPLITS algorithm attempts to break these into their individual components by examining successively higher threshholds above the detection isophote. The threshhold
step is some fraction of the pixel-to-pixel standard deviation (default is 0.2). SPLITS
works on a rectangle of pixels encompassing the detection isophote. Like DETECT,
it searches for pixels above a given threshhold but it uses the raw image pixel values,
not the filtered values. Again, a minimum number of pixels are necessary to define
an object. New objects are identified as isolated groups of contiguous pixels above
each threshhold. Ideally, the algorithm would split the merged isophotes of galaxies
without producing spurious objects in the diffraction spikes of stars and extended
halos of galaxies. Unhappily, this is usually not the case without great care. The
only explicit control of the SPLITS routine is achieved by adjusting the zero-point
and slope parameters that the algorithm uses to estimate the pixel-to-pixel noise from
the pixel values. Increasing either of these will suppress splitting.
A point-spread function is automatically constructed for each image to distinguish between stellar-like and extended objects (i.e., galaxies). The PSF template is
found using the AUTOPSF routine in FOCAS. This algorithm corrals a minimum of
stars with a narrow dispersion in the radial moment of their light distribution and averages them to construct the template. Since the object catalogs are constructed from
relatively deep images they are dominated by galaxies at the faint limit (R , 23),
which may interfere with the construction of a true stellar template. I avoid this problem by including only objects brighter than R < 21. Very bright objects (R < 16)
whose images may be influenced by the non-linear response regime of the CCD and
are also excluded.
FOCAS separates objects into classes using a series of rules and a "resolution classifier scheme" which tests the surface brightness profiles of objects against
templates constructed from the image point-spread function and broadened versions
thereof. Objects which contain saturated pixels (saturation level set by the user)
are automatically classified as stars. Objects with asymmetries or ellipticities greater
than 0.9 are classified "long" (e.g. charged particle events). Objects which cannot be
successfully modeled as a combination of normal stellar profile and a stellar profile
broadened by less than a factor 9.5 are considered galaxies.
I use the standard classifier parameters to a magnitude of R < 21 to separate
stars and galaxies.
Beyond this point in a typical image the classifier begins to
fail and galaxies become mistaken for stars, as evidenced by an enormous rise in
the differential "star" counts by R - 21.
The galaxy counts become sensitive to
atmospheric seeing and errors are introduced in the cluster/background differencing.
Fortunately, the surface density of stars per unit magnitude at high galactic latitude
begins to level out at R - 19 and is much less than the number counts of galaxies at
levels two magnitudes fainter. In Fig. 6 I compare the differential number counts of
stars with those of galaxies derived below. The counts are roughly equal by R - 19.
To extrapolate the bright star counts to fainter levels I use the model of Bahcall &
Soneira (1981), choosing to renormalize using the numbers of bright stars identified
in the images. Galactic coordinates are taken to be b = 301, 1 = 0, and the dominant
population at these faint magnitudes is assumed to be low-mass red disk stars with
V - R-1. By R - 21 the galaxy counts are a factor of 6 higher than stars. The
extrapolated model indicates that the contribution of stars at fainter magnitudes is
small and thus I assume all objects at R > 21 classified as either stars or galaxies are
actually galaxies.
FOCAS calculates a number of photometry parameters; only those used in
the analysis are discussed here. The isophotal area is simply the number of pixels
above detection threshhold associated with the object. An isophotal magnitude is
calculated from these pixels using a zero-point supplied by the user. First and second
moments of the x, y and radial coordinates are computed, both for the unweighted
and intensity-weighted distribution of pixels in the detection area. A total magnitude
is calculated by filling in concavities in the isophote shape and growing rings around
the object until the area exceeds the isophotal area by a specified ratio. The total
magnitude calculation is often unreliable, failing or having errors in crowded fields.
The isophotal magnitude is used here instead.
Each catalog generated from each field was individually inspected for problems
such as a very asymmetrical or noisy point-spread function template; spurious faint
objects generated in artifacts such as the diffraction spikes of stars; or erroneous
splitting of large, extended objects into smaller fainter ones. Very often, adjustments
were necessary to the point-spread function generation or splitting machinery to fix
these difficulties.
2.1.6
MAGNITUDE ERRORS
Errors in galaxy photometry are usually dominated by systematic effects such as
poor isophote definition and the finite size of pixels rather than the Poisson noise
of the total counts. I estimate magnitude errors by using cluster galaxies that were
multiply imaged in the CCD image mosaic of Abell 1632. A plot of the errors vs.
magnitude is shown in Fig. 2. The RMS error in the isophotal magnitude was found
to scale with apparent magnitude as
1 0 m/5,
i.e., Poisson-like behavior. However the
errors are much larger; of order 5% for the brightest galaxies.
The zero-point of
the magnitude-dependence is assumed to be the rest-frame surface brightness of the
detection isophote plus a constant. I find,
or(m) = o0-0
- ( " ° - 5 .0- m ) / 5
(2.8)
where ao = 0.10 magnitudes.
2.1.7
COMPLETENESS
In the game of galaxy counting it is important to make corrections for, or at least
estimate the completeness of a galaxy catalog at a given magnitude, i.e., the ratio
of the number of galaxies that are detected to the actual number. In this project,
completeness corrections become particularly crucial when estimating the relative
numbers of low-luminosity galaxies in clusters. For the apparent magnitude range of
interest R - 21 - 23, the background field galaxy counts are comparable to, or often
greater than the expected cluster counts. Thus small changes in the completeness
estimate can have a large effect on the result (a classic case of the peril of subtracting
two large numbers to find a small one). Completeness also depends on characteristics
other than the isophotal magnitude (e.g., surface brightness): To construct cluster
LFs, galaxies are ordered only by apparent magnitude and the completeness at a
given magnitude is the integrated completeness over all other galaxy properties.
Most of the incompleteness in an imaging survey arises from three effects.
First, galaxies with true apparent magnitudes near the magnitude or surface-brightness
limits of the survey can fall below the detection limit due to the superposition of sky
noise, errors in the CCD response, etc. Second, galaxies can be detected but misclassified as other objects. For example, compact dwarf elliptical (dE) galaxies in distant
clusters might be mistaken for stars (Ferguson & Binggeli 1994). Third, faint galaxies
may fall near bright stars or galaxies and not be discerned as separate objects by the
detection algorithm. (In the reverse reaction, close pairs or multiples of stars may be
mistaken for galaxies).
The last effect is a purely random phenomenon and can be robustly estimated
through the use of Monte Carlo simulations of image. However, the first two effects
depend both on random events (like a star being on the line of sight to a galaxy),
and systematic dependences on the detailed properties of the galaxies. Without a
priori knowledge of the full spectrum of galaxy luminosities and surface-brightness
morphologies it is impossible to estimate the true incompleteness. For example, there
may exist a host of extended, very low surface brightness galaxies in clusters which
would never be detected in this survey.
One approach to this problem is to postulate a variety of galaxy morphologies
to construct artificial galaxies images at a fixed magnitude which are then added
randomly to the real images.
The fraction of galaxies which are recovered is the
completeness estimate. For a detailed description of this technique I refer the reader
to Steidel & Hamilton (1993). The one disadvantage of this method is that without
deeper, higher-resolution image one is never certain that the galaxy models correspond
to reality. In the case of galaxy clusters this is particularly troublesome since the
morphologies of the dwarf population in clusters are poorly characterized, and may
even vary from cluster to cluster.
An alternative method is to assume that Nature is kind and that the galaxies
that are detected at a given apparent magnitude span the full range of morphologies,
although the distribution may not be representative due to the selection effects dis-
cussed above. A large number of faint objects can be combined to construct a high
signal-to-noise galaxy template which is representative of galaxies at that magnitude.
This template is then added repeatedly to the real images and the fraction which
are recovered in the catalogs is found. Although the details of the individual galaxies are lost in the averaging process, the important gross properties, i.e., the radial
distribution of surface brightness, are retained.
Of course, by using those galaxies which are already detected to make the
template, one biases the completeness estimate towards a kind of upper limit. There
may be additional objects, lurking in the noise, which are not detected and do not
contribute to the template. A less biased template could be made by weighting the individual contributions by some a prioriestimate of their detectability. With enough
galaxies, in principle one could recover a very acccurate estimate of the completeness through interleaved iteration of the galaxy detection and template construction
processes.
At a given isophotal magnitude, the 50 galaxies with the closest magnitude
are selected and 10 x 10 arc-second sections centered on each of these galaxies are
extracted from the image being considered. These are averaged, with exclusion of
the three highest and the lowest pixel values at each pixel to eliminate neighboring
objects or image defects. The template is then added to 100 random locations in
the image and the fraction of recovered objects recorded. An object is considered
recovered if an object of similar magnitude is found within 0.7" of where the artificial
galaxy was placed. (The probablity of a spurious match is of order 2%).
2.2
FIELD GALAXY COUNTS
Accurate estimation of the contribution of field galaxies (mostly background galaxies)
to the counts in cluster images are critical in determining the shape of the luminosity
function, particularly its faint end, and in estimating the luminosity overdensity of the
cluster within some aperture. Errors associated with statistical background galaxy
subtraction are usually the limiting factor in detailed cluster galaxy measurements.
It is also important to estimate the field-to-field variance in the counts in order to
calculate the errors in the cluster LF. My approach here is to fit a model to the counts
and describe the fluctuations as the variance in the model parameters.
To obtain accurate estimates of field galaxy counts at faint magnitudes I used
a series of nine very deep images obtained with a thinned 20482 Tektronix CCD with
a Rs filter at the prime focus of the Kitt Peak Mayall 4-meter telescope and kindly
provided by C. Steidel. The CCD was operated in 2 x 2 binning model with a final
pixel scale of 0.48 arc-seconds and a field of view of 10 arc-minutes. These images were
obtained as part of a program to find the galaxy counterparts to z
-
1 absorption
line systems detected in the spectra of z - 2 QSOs (Steidel & Hamilton 1993) and
thus, for my purposes, are essentially random fields. Details regarding the images
are in Table 4. The basic image processing and photometry were already performed
by C. Steidel and collaborators.
Some further image processing was done; Bright
stars and charge-transfer tails were masked out and replaced with uniform sky value
and in some cases large-scale variations in background removed using cubic spline
fits to the surface with one or more segments and iterative 3o pixel value rejection.
All but one of the images were brought to a common lo- surface brightness limit of
28.05 magnitudes arc-sec. - 2 or an effective detection limit of PR, = 25.63, about 1.5
magnitudes deeper than the cluster survey images. This is accomplished by simply
adding random Gaussian noise to the images in appropriate amounts. The remaining
shallower image has a puR, limit of 25.23. The limiting magnitude is R - 24.5 although
the incompleteness becomes appreciable two magnitudes brighter.
Galaxy catalogs were produced using the FOCAS analysis procedures described above. Apparent magnitudes were transformed to Kron-Cousins R assuming R, - R-0.14 (Steidel & Hamilton 1993). A Galactic extinction correction was
applied using the AB values in the NASA Extragalactic Database and assuming
AR/AB = 0.63. Figure 3 plot the differential galaxy counts for the summed cata-
logs in the magnitude range 17 < R, < 24. The spatial detection filter optimized for
clusters at z = 0.175 (an intermediate value in the survey) was usd. The data here
is binned in 0.25 magnitude intervals; solid error bars are the errors in the weighted
mean of the fields; the dashed error bars represent the RMS field-to-field scatter.
There is a break in the counts at R - 21 where the slope of the number counts
flattens.
To accurately model the number counts, the intrinsic fluctuations in the background must be distinguished from systematic errors. Several systematic effects will
cause errors in the field galaxy counts. Here I consider three such effects. Because the
galaxy photometry is performed using limiting isophotes, Galactic extinction will not
uniformly fade all objects, but attenuate those with the lowest surface brightness the
most. In the extreme case of low surface objects near the detection limit, the fading
can remove them from the catalog althogether. Atmopheric seeing will blur the images of stars and galaxies, making them more difficult to detect and separate. Finally,
the choice of the angular size of the detection filter, optimized for the detection of
closer cluster galaxies, may have a significant impact on the counts. These should
be considered separately from the intrinsic fluctuations and random errors since their
effects can in principle be measured and accounted for.
Two independent, non-parameteric tests for significant deviant behavior of
the number counts in a field are the total counts within the magnitude range of
interest and a Kolmogorov-Smirnov test of the apparent magnitude distribution. The
latter is sensitive only to the distribution of values over some interval, not the total
number of values. I examine compare the counts in the apparent magnitude range
17 < R < 24 of each field to that of the summed fields. The bright limit is the
brightest magnitude where there is still a significant number of galaxies; the faint end
is where incompleteness of the catalogs becomes appreciable. I find that the K-S test
is unable to distinguish between any of the individual fields and the average. The
total galaxy surface density of each field is reported in Table 4.
The 0421+019 field counts are lower by a factor of 3.6 over the scatter of the
remaining fields. This field has the highest Galactic extinction and a plot of surface
density versus AB shows the expected strong correlation (Fig. 4). With it are plotted
the number counts in the 0421+019 field as a fraction of the average of five fields
with (AB) = 0.01, after the magnitudes are corrected for extinction. The relative
counts deficiency in the former field grows with fainter magnnitude. After correcting
the 0421+019 for an estimated 0.25 magnitudes extinction in the R, band I find that
the counts become deficient fainter than R = 20, falling to a relative completeness
of 60% by R = 24.
This means that in addition to fading all the galaxies, the
surface brightness effect of extinction causes a fraction of the faint galaxies to drop
out of the catalog. I adopt a crude empirical description of the combined fading and
incompleteness: The surface densities are adjusted using the empirical relation
log, = logN(o -0.63(y
+ 0.47)AB
(2.9)
After this correction I look for a correlation with seeing (here described as the
intensity-weight radial moment of the PSF); None is seen (Fig. 5).
The width of the spatial detection filter in FOCAS is varied inversely with
cluster redshift to maintain a constant linear scale at the distance of the cluster
galaxies. A possible systematic effect on the counts of background galaxies (which
have more or less a fixed redshift distribution) was investigated by analyzing the field
images using detection filters of different widths. The number counts from one of
the fields are shown in Fig. 5 for filters optimized at differrent redshifts. There is a
significant suppression of the galaxy counts fainter than R = 21 for filters optimized to
redshifts z < 0.10. The effect is dramatic; for z = 0.1 filters the counts are consistent
within the errors with higher-redshift filter; at z = 0.09 the counts are lower by a
factor of about 2 to R < 24. Thus in the analysis of the cluster surveys, the widest
detection filters used are optimized for z = 0.1.
The catalogs from eight image are combined (the high extinction field is excluded as a conservative measure) to construct a model of the mean number counts.
The magnitudes in each field are corrected by 0.63AB magnitudes of extinction and
the associated losses due to extinction are corrected by multiplying the field of view
by the factor
.
Completeness corrections (never exceeding 10% for R < 24)
are made using the procedure described in §2.1.7. I fit a broken power-law normalized
10
-0.30AB
at R = 21.0 to the number counts in the range 17 < R < 24;
n(17.0 < R < 21)
= n21 10 -Y(R-21)
n(21 < R < 24) = n21 1 0Y2(R-21)
(2.10)
(2.11)
The slopes 'i, '2 and associated la errors are found by maximum likelihood analysis
of the data and Monte Carlo simulations thereof. The normalization at R, = 21.5
is set to match the total number of counts within the magnitude range. I find -Yi =
0.505 ± 0.014, /2 = 0.304 ± 0.006 and n 21 = 4472 ± 120. The solution is plotted as a
solid line in Fig. 3. The formal X2 for 0.5-magnitude bins (14 degrees of freedom) is
11.9 based on Poisson errors. This relatively high value of X2 is not surprising given
that the fluctuations in galaxy counts are super-Poissonian (see below).
Several other results are plotted with the counts; two fields with the same
Rf filter (Steidel & Hamilton 1993); deep R-band counts using the Keck 10-meter
telescope (Smail et al. 1995); the CCD imaging of Metcalfe et. al. (1991) taking
R = rF + 0.13; and the rF Digitized Palomar Sky Survey II photograph counts which
are well described by loglo n(rF)dm-0.327rF- 3.20 (Weir et al. 1995). There is some
disagreement between the counts, but they bracket the results derived here.
My
counts are lower than one of the Steidel fields and the Keck fields but are in turn
significantly higher than the counts of Metcalf et. al. at R - 22 and nearly identical
with the R < 20 counts of Weir.
As mentioned previously, a proper analysis of cluster luminosity functions requires one to estimate the errors associated with the LF parameters. A major source
of error arise in the process of statistically subtracting the background galaxies, which
are determined from independent fields. Field-to-field fluctuations in the background
galaxy counts will introduce uncertainties into the resulting LF. The errors can be
estimated using Monte Carlo simulations of the data where the Gaussian-distrubted
variation isintroduced into the background parameters. It is observationally wellestablished that, due to the clustering of galaxies on the sky, the field-to-field variance in the number counts exceeds the Poisson estimates by a significant factor,
typically 2-3 (Jones et al. 1991). The variance depends both on the angular field of
view and, through the amplitude of the galaxy-galaxy correlation, on the magnitude,
with fainter galaxies being less strongly clustered (Efstathiou et al. 1991). If the
angular correlation function of galaxies to some magnitude limit obeys w(O) ~ 0-0. s
(Peebles 1993) then the field-to-field variance in the number counts will be,
S(N -
2=
N2)
N + C(m)O-1 8
(2.12)
where 0 is now the field of view and C describes the strength of the clustering and is
a function of the magnitude range.
To measure the effect of this scatter on the description of the backgound counts
used here, the broken power-law model is fit to the eight individual 10 x 10 arc-minute
fields. The background fluctuations will manifest themselves as field-to-field variance
in the model parameters. In addition, single 5 x 5 arc-minute subsections of each field
are then extracted, and with a seventh small field, are analyzed in the same way to
examine the dependence on the angular field of view.
As an important aside, I find a significant covariance between the faint-end
slope and the R = 21 normalization in the sense that the variance in the faint galaxy
counts is minimized (Fig. 6(b)).
If the counts in the range 21 < R < 24 were
constant, then the variation in n 21 and 72 (Sn 21 and Sy2) are related by,
•n 2 1
n21
(1
72
_ 37 ln(10)103 12 )
10 3 "Y2 - 1
(2.13)
/
For 72 = 0.304, Slogn 21 '-0.90&7
2 . This locus is plotted in Fig. 6(b). The close
correspondence with the data motivates fixing 72 to maintain a constant number of
galaxies while the normalization at R = 21 varies.
For the 100 square arc-minute fields the field-to-field RMS scatter in the bright
slope "yi is 0.08 about a mean of 0.40. The variance in n 21 is 19%, or 2.4 times the
Poisson value. For the 25 square-arc minute fields the, r,, = 0.14 about a mean of
0.45, and u, 21 = 20% (1.44 times the Poisson value). These are consistent with the
results of Jones et. al. (1991)
2.3
GLOBAL GALAXY CLUSTER PROPERTIES
If either the formation or subsequent evolution of galaxies is significantly affected
by the cluster environment, or by its humble beginnings as a fluctuation on a mass
scale of 1015AM®, this phenonomen might manifest itself by systematic correlations
of galaxy properties with the global properties of the cluster. The two traditional
properties that have been investigated are the richness and the morphology of a
cluster. The former refers to the projected density of galaxies in excess of a mean
background, usually in a fixed metric aperture. The second describes the geometry
or shape of the cluster galaxies or its graviational potential. Cluster classifications
based on subjective inspection of photographic plates exist in the literature (Bautz
& Morgan 1970; Rood & Sastry 1971) but lack any quantitiative or objective nature.
Most of the clusters in my sample are also too distant for photographic plate material.
Thus I attempt some kind of classification using the information in the limited field
of view of the CCD imaging.
2.3.1
CLUSTER RICHNESS
Abell (Abell 1958) counted the number of galaxies CA in excess of the background
level within an aperture of radius 1.5 h -
1
Mpc and with magnitudes between m 3
and m 3 + 2, where m3 is the magnitude of the third brightest galaxy in the cluster.
He assigned richness classes, with R = 0 clusters having 30-49 galaxies, R = 1 with
50-79 galaxies, R = 2 with 80-129, R = 3 with 130-199 galaxies, R = 4 with 200'299 galaxies, and the single R = 5 cluster in the catalog, Abell 665, having at least
300 galaxies. The counts and assigned classes are generally found to be accurate to
about one richness class, with a systematic error of underestimating the richness with
increasing redshift. As an extreme example of error, the very rich (and most distant
at z = 0.370) cluster Abell 370 was assigned a richness class of 0 in the catalog.
I make an independent measurement of the cluster richness with which to
correlate galaxy properties. Rather than use galaxy counts, I estimate the luminosity
overdensity (with respect to a estimated cosmological mean value) within an aperture
centered on the peak of the cluster surface brightness. Luminosity density may be a
more robust measurement than counts for two reasons: It is not dominated by the
faintest and most numerous galaxies and therefore not sensitive to the exact cut-off
in absolute magnitude; Second, merging of galaxies or 'cannibalism' by the brightest
cluster galaxy will preserve luminosity but not number.
I calculate the luminosity overdensity SL within a fixed projected linear distance from the cluster center. I identify the cluster center as the peak in the surface
brightness distribution of the cluster, measured with some smaller aperture. The
result may depend on the size of the smaller aperture used, and thus I do not wish
to make an ad hoc choice. Instead, I define an aperture which is appropriate to the
mass scale of a Brightest Cluster Galaxy. To do this, I use a standard BCG luminosity and calculate a mass using a M/L value typical of giant galaxies. I then assume
that the density profile of the cluster is that of an isothermal sphere and that the
Abell radius of the cluster corresponds to the virial radius where p = 178p. This
choice of profile and normalization give the radius containing the BCG mass scale.
Taking MBCG = -23.60 + 5log h (see 2.5) and M® = +4.3 in the R-band to find
LBCG = 1.45 x 101 h- 2 L®, and assuming a mass-to-light ratio of 100h(M/L)®, this
mass scale is 1.5 x 1013h- 1 M®. Taking the virial radius of clusters at the present
epoch to be the Abell radius rA = 1.5h - 1 Mpc then the mass contained within rA is
roughly
MA = (
) (178
),
(2.14)
or 3.8 x 1014 h - 1 M® (assuming fQ = 0.2). For an isothermal sphere, the mass contained
within a given radius is proportional to that radius. Thus the radius containing an
equivalent BCG mass is then 0.04 of the Abell radius or 55 h - 1 kpc. This value is
comparable to what is often taken to be the "core radius" of clusters. I define the
corresponding angle for the BCG assuming a qo = 0.1 cosmology;
0.04 (1 + z) 2
sBCG = 0.1z - 0.9 (V1 + 0.2z - 1)
in seconds of arc.
(2.15)
I use the LCRS estimate for the mean luminosity density of the Universe
(Lin et al. 1996) and thus for consistency I calculate the total luminosity from all
MR
< -17.41
galaxies within a radius of 0.5h -1 Mpc. The flux from all galaxies
within the corresponding angular distance is summed and corrected for missing and
excised areas outside the CCD field of view assuming a 0-
1
projected surface density
profile (Beers & Tonry 1986). The contribution from the background is subtracted
and the remaining flux is converted to luminosity using the redshift of the cluster.
This luminosity is to be compared with the cosmological mean luminosity with
a sphere of the same radius in order to calculate a luminosity over-density 3 L. To do
this, a correction must be applied the cluster luminosity to remove the contribution
from outer regions of the cluster projected along the line of sight. To estimate the
magnitude of this effect, I consider a 'toy' model of a spherically cluster with a powerlaw luminosity density profile. The ratio of the luminosity within the sphere compared
to the total projected luminosity along the line of sight of the sphere is,
F=
(3 + n) fo In (1 + z - 2 ) dz'
(2.16)
(2.16)
where n is the power-law index. Taking n = -2, I find F = 2/r. I use the luminosity
density estimated from the Las Campanas Redshift Survey over the same absolute
magnitude range; PL = 1.9 x 10hLe®Mpc- 3 . The mean luminosity within a sphere
of 0.5 h -
1
Mpc is 1.0 x 108 h- 2 L® for MR < -17.41
+ 51og h. The random error
in the overdensity is taken to be the Poisson fluctuations associated with the finite
galaxy counts within the aperture. I ignore the error in the calculation of the mean
luminosity density since this will only produce a systematic offsets in all cluster values.
Note that SL does not depend on h or Q.
I calculate SL for all clusters, both photometric and non-photometric, keeping
in mind that non-photometric conditions will systematically decrease estimates of SL.
Fig. 7 is a plot of SL versus redshift. There does not appear to be a significant trend
with redshift.
A comparison between CA and SL for the photometric clusters appears to be
simply a scatter plot (Fig. 7). The "sore thumb" is Abell 2218, the richest cluster
in the survey. Even if the two quantities are uncorrelated, are they consistent? I
compare the overdensity computed (or estimated) within the 0.5 h - ' Mpc diameter
aperture with the Abell counts. To do this, I convert CA into an equivalent SL. I
assume that the PL - r- 2 profile extends at least to the Abell radius of 1.5 h - 1 Mpc,
such that the overdensity within the Abell radius is SA = J0.5/9- I then remove the
correction of 2/7 made earlier to remove outer cluster galaxies projected onto the
field of the inner cluster. To convert to galaxy counts, I assume a Schechter function
with a flat faint-end slope a (the exact value is not important here since the counts
do not extent much past M*). I assume that the third-ranked galaxy is actually the
second brightest galaxy that is drawn from the Schechter LF and use the following
definition,
CA =
3+2 exp (_10-0.4(m-*))
0
1.5 = o
J
exp
(-10-0.4(m-m*))
dm
dm
(2.17)
(2.18)
For each value of CA the value of m 3 - m* is found by evaluating the ratio of the
two equations, whereupon the normalization 40 is determined. The total luminosity
within the Abell sphere to MR = -17.41
(about M* + 2.8) is then estimated and
divided by the mean field value (0.58 L* per Abell volume);
2
S
7
1
4
4.64m
m3+2
0
m+2 exp (10-0.4(m-m*)100.4(m-m*)) dm
(2.19)
3
Interestingly, CA = 100 gives an overdensity of
6
A
= 170 within the Abell volume,
consistent with the Abell radius being the approximate virial radius of a rich cluster.
Finally SA is divided by nine to give SL within the 0.5 h - 1 Mpc radius. The result is
plotted in Fig. 7. The good agreement is comforting, if not terribly impressive due
to the large scatter in the data. I thus conclude that my overdensity estimates and
Abell's richness classifications are consistent.
2.3.2
MORPHOLOGY
Abell established a limited morphology classification for his clusters; However, the
two better-known morphological classifications for clusters of galaxies are the BautzMorgan and Rood-Sastry types. The classification of Bautz & Morgan (1970) is
based upon the luminosity of the brightest cluster galaxy with respect to the other
cluster members. Type I clusters contain a central cD galaxy; Type II contain a giant
elliptical or ellipticals, and Type III contain no dominant galaxies. Standard examples
given by Bautz and Morgan for the three types are Abell clusters 2199 and 2029 (I);
Abell clusters 194, 1656 (Coma), and 2197 (II); and the Virgo and Corona Borealis
(Abell 2065) clusters (III). The morphological classification of Rood & Sastry (1971)
is based more upon the spatial distribution of the brightest galaxies and is arranged
into a "tuning fork" with two branches. The base of the fork is occupied by cD
clusters with a central supergiant galaxy. At the branching point are B (binary)
clusters with two giant galaxies of roughly equal luminosity. One branch contains L
(,linear) and F (flattened) clusters with a high flattened distribution of bright galaxies,
while the other contains C (core-halo) and I (irregular) clusters with a more circular
or irregular distribution. Bautz & Morgan suggested that this morphological sequence
might correspond to an evolutionary sequence from older to younger clusters.
The optical characterization of galaxy clusters since that time has concentrated
on quantifying the amount of so-called "sub-structure" in the spatial and velocity
distribution of galaxies. Older, dynamically relaxed clusters are expected to exhibit
less; dynamically young, more. A complete review of the subject is beyond the scope
of this thesis, particularly since most of the techniques are unsuitable for this data.
The reader is referred to a review and comparison of different techniques in Pinkney
et. al. (1996). Substructure tests have also been developed and applied to the surface
brightness morphologies of the X-ray emission which originates from an intracluster
medium in approximate hydrostatic equilibrium with the gravitational potential of
the cluster (Mohr et al. 1995; West et al. 1995). In particular Buote & Tsai (1995;
1996) have pursued a fundamental approach, performing a multipole expansion of the
graviational potential and constraining the relative "power" in each moment using
X-ray measurements. Rhee et. al. (1992) and Pinckney et. al. (1996) perform a
somewhat similar, but less physically based analysis of optical data using a Fourier
expansion of the galaxy distribution.
Although it is tempting to perform a multipole expansion here using galaxies
in place of X-ray photons, there are several reasons why this is impractical with
this data. First, the distribution of galaxies on - 1 Mpc scales is much less centrally
concentrated than the X-ray surface brightness: The data indicates that the projected
surface brightness profile of Abell clusters falls as 0 - 1 . The X-ray emissivity of the
intracluster medium, on the other hand, is proportional to the square of the gas
density and the X-ray surface brightness is observed to fall off with a much steeper
index of -3 to -4 (Jones & Forman 1984). A flatter profile leads to much larger random
errors in the moments. Secondly, the number of independent optical measurements
(locations of bright galaxies) is of order 102 - 103, an order of magnitude smaller than
the number of X-ray photons from a cluster observed with a reasonable integration
time, and also leading to larger random errors. Finally, the finite optical field of
view and significant offsets between the true center of the cluster and the coordinates
given in the Abell catalog (determined from the projected overdensity on a much
larger scale of 3 h - 1 Mpc) lead to spurious contributions to the moment calculations
which cannot be removed without making dubious assumptions about the symmetry
of the cluster geometry. Finally, bright stars in the cluster foreground can null sections
of the effective field of view, producing erroneous moments.
Instead, I attempt to classify the morphologies using two parameters that
describe (1) the concentration of galaxies towards the cluster center and (2) the
relative prominence of the cluster center with respect to other peaks in the surface
brightness.
For the first parameter, I return to a comparison between the Abell counts
within 1.5 h - 1 Mpc and overdensity computed within 0.5 h - 1 Mpc. Some of the
scatter is undoubtably due to Poisson and photometry errors (Abell's as well as
mine). However, 1 magnitude of error moves the points only 0.4 dex and probability
will not qualitatively change the general distribution of the points. An interesting
hypothesis to explore is that the Abell counts are accurate to the limit of the Poisson
statistics and that the observed large scatter between the overdensity and CA is due
in part to variations in the radial profile or morphology of the cluster. Dynamically
relaxed clusters with highly dense cores would have much larger values of S than
similarly rich but less concentrated clusters. If this could be shown to be the case the
ratio of the two values would be a important quantity with which to classify galaxy
clusters.
Fig. 8 is a somewhat complicated plot which encodes the the standard morphological classifications in the points (see §2.3.2). The points are larger with lower
numbered Bautz-Morgan type, and become rounder with later Rood-Sastry types.
If these classification correspond to dynamical ages, the larger, rounder points are
older and thus might be exptected to show a greater concentration of galaxies in their
cores, increasing SL with respect to CA. The plot does suggest such a correlation.
Motivated by this possibility, I define a concentration index
Ic = loglo
(2.20)
.
which takes on a zero value for an isothermal sphere.
the procedure described above. Abell counts,
60.5s,
SA
is calculated according to
and Ics for all clusters are given
in Table 3.
The second parameter, a "peak index" Ip, is defined as
= -logl
Ip
0
()
(2.21)
where fi and f2 are the fluxes of the brightest and second-brightest peaks measured
in the 0.055h -' 1 kpc aperture defined in §2.3.1. The second-ranked peak is found in
the same way as the first-ranked, with the added requirements that the peak must be
a true peak, i.e. all neighboring grid values must be smaller, and its center must be
separated from the center of the first-ranked peak by at least 4 aperture radii (220h-'
kpc).
A cluster characterized by a single very bright galaxy or concentration of
galaxies (Rood-Sastry types D and C) would have a large value of Ip, while a bimodal
cluster or irregular cluster would have a value approaching zero. I expect a redshift
bias in that the brighter second-ranked peaks of nearby clusters will be more likely
to fall outside the field of view and P1 will tend to decrease with redshift. This trend
is shown in Fig. 8.
Finally, I plot the clusters in Ic - Ip space (Fig. 9) with the points coded
with Bautz-Morgan and Rood-Sastry type as before. The arrow shows the qualitative
direction in which clusters are expected to evolve as they dynamically relax to a more
concentrated state with a central peak. There is a tendency for the earlier (more
relaxed) Bautz-Morgan and Rood-Sastry types to fall in the region of higher Ip and
Ic but there is considerable scatter.
2.4
2.4.1
LUMINOSITY FUNCTIONS
LF CONSTRUCTION
The galaxy catalogs constructed from imaging of galaxy clusters at known redshifts
can be used to construct luminosity functions, subject to the intrinsic photometric
errors and uncertainties in background galaxy subtraction. A composite cluster LF
can be used to determine the mean faint-end slope and the relative abundance of
dwarf galaxies in clusters.
A comparison of luminosity functions from cluster to
cluster, and between clusters and field, can be used to ascertain whether galaxies
obey a "universal" luminosity function or if different environments have produced or
evolved different galaxies.
The analytical form of the LF proposed by Schechter (1976) and motivated by
the Press-Schechter formalism (Press & Schechter 1974) has been repeatedly shown
to provide a fair description of both cluster and field galaxies, provided one allows
the two free parameters M* and a to vary. I thus use this convenient formulation to
describe my data and compare it with previous work.
The absolute magnitude limit to which galaxies are included in fitting an LF
model must be chosen with some care. Evidence for a steep upturn in the LF at faint
absolute magnitudes indicates that the results of a LF fit may depend sensitively on
the elected cut-off. Indeed, some of the disagreement between different previous work
may have arisen from differences in the chosen limits in M. To be consistent with the
analysis of the Las Campanas Redshift Survey (Lin et al. 1996), I use only galaxies
brighter than Mr = -17.5 + 5 log h (MR = -17.41)
unless otherwise mentioned. I
also exclude the first-ranked, and sometimes the second-ranked, galaxies since these
do not appear to obey statistics commensurate with a Schechter function (see §2.5).
By far the most critical process in constructing a luminosity function is correct subtraction of the background field galaxy contribution. Modest errors in the
background counts can produce substantial errors in the derived LF. I use two independent and complementary techniques to accomplish this. The first technique uses
separate, random fields to establish the background counts (§2.2). While being a
more objective method the results can suffer from random errors due to the fluctations in the background counts as well as serious sytematic errors including errors
Iin photometry between the cluster and background fields, differences in seeing, and
different completeness corrections. The second technique differences the high-density
and low-density regions on the cluster image(s) to remove the background (as well
as some of the cluster signal). This generates results that are free of the systematics
described above, but assumes that the luminosity function of cluster galaxies is independent of the local density. Since the morphological breakdown of galaxies depends
on the local density (Dressler 1980), there is no reason to suppose that the luminosity
is invariant. Differencing high- and low-density regions may result in a LF which is
skewed from the true LF in either region. However, performing both operations on
the same data set and the results checked against each other is potentially a powerful
technique to isolate such errors, and this is the approach I take here. Below I discuss
some mechanical details relevant to each method.
Absolute background subtraction: A model consisting of the background
model constructed in §2.2 and a Schechter function (modulus the redshift of the
cluster) is convolved with an estimate of the completeness and fit to the distribution
of galaxies with apparent magnitude to find the two free parameters M* and a.
A typical cluster catalog generated from CCD imaging of clusters contains a few
hundred bright galaxies, only a small fraction of which will be brighter than M*.
In this low-signal regime it is clearly hazardous to pursue the traditional route of
sorting the galaxies into bins with magnitude and attempting a least-squares fit of
a Schechter function to the count; the exact choice of bins may significantly effect
the value of M* obtained, and thus a.
Instead, I use the method of maximum
likelihood [e.g., (Efstathiou et al. 1988)] which treats the data in unbinned form.
A galaxy count model is constructed from a Schechter function superposed on the
power-law background count model described in §2.2. The two free parameters of
the Schechter function are determined through the method of maximum likelihood
(the LF normalization is fixed to match the total number of galaxies). A "black-box"
extrema-finding algorithm, kindly provided by J. Tonry, was used.
Random errors in the Schechter function parameters are produced by (1) shot
noise from the finite number of galaxies used to construct the luminosity function; (2)
galaxy magnitude errors; (3) intrinsic fluctuations in the normalization and slope of
the background counts; and (4) photometry errors from uncertainties in atmospheric
and interstellar extinction. I estimate errors by Monte Carlo simulations of the data
using the following prescriptions: The magnitudes of the simulated galaxies are randomly drawn with replacement from the real data set. The total number of galaxies
is held fixed since this is an independently measured quantity. Gaussian-distributed
errors in isophotal magnitude are introduced using the relation described in §. The
normalization and slope of the background counts are varied by a Gaussian with the
standard deviation given by Eqn. 2.8. Although the variations in these quantities
are covariant, as a simple, conservative approach I vary them independently. Finally, I add a Gaussian-distributed error of 0.03 magnitudes in the offset between the
background and cluster photometry to account for possible errors. This accounts for
maximal errors of 0.02 magnitudes in atmospheric extinction and 0.02 magnitudes
in interstellar extinction. I estimate the lcr errors in the parameters using the 68%
percentile limits of LF parameters from 100 Monte Carlo simulations. I then calculate
a goodness-of-fit by calculating the X2 per degree of freedom between the data binned
in half-magnitude or magnitude intervals and the best-fit luminosity function.
Differential Background Subtraction: This is a procedurally much simpler technique. The center of the cluster is identified as the peak in the cluster surface
brightness using a 0.055 h-
1
kpc aperture, as described in §2.3.2. Galaxies are then
separated into inner and outer regions, depending on their projected distance from
the cluster center. Typically a separating radius of 0.5 h- 1 Mpc is used. The separate catalogs are then binned in magnitude, and differenced, weighting the bins by
the inverse of the accessible area on the sky. A completeness correction is then made.
I require that the outside region have an area at least 1/3 of the inner region to have
a sufficiently accurate estimate of the "background".
2.4.2
COMPOSITE LUMINOSITY FUNCTION
A composite LF is constructed by compiling the galaxy catalogs from 22 clusters imaged with the Loral CCD/R, filter combination under photometric conditions. There
are 5489 galaxies in the range -23.41
second-ranked galaxies).
< MR < -17.41
(excluding the first- and
The best-fit Schechter function using the absolute back-
ground correction has Mý = -21.13 + 0.11 + 5log h and a = 1.09 ± 0.08. This is
plotted in Fig. 10 with the background-subtracted and completion corrected data
binned into 0.5-magnitude bins (solid points). The catalog is essentially complete to
an absolute magnitude of-17, but is only 50% complete at MR = -15.5.
X2
The reduced
(14 degrees of freedom) is 14.2, indicating there is considerable structure that is
not well described by the simple choice of LFs. The most noteworthy of these is a dip
at MR- - 19.7. The distribution of parameters with 1000 Monte Carlo simulations
of the data are plotted with the lao limits in Fig. 11. If I include the second-ranked
galaxy the parameters changes in value to MR = -21.30 and a = 1.13, a difference
that is not significant compared to the errors in the values.
Much previous work on cluster LFs with shallower photographic data has fixed
the faint-end slope to a = 5/4, as prescribed by Schechter (1976), to obtain a welldefined value of M.. The two Schechter parameters are strongly covariant; thus to
compare with this work I also fix a to this canonical value, to find M7 = -21.47±0.14.
The difference in predicted counts between the two fits is no more than 25% over the
magnitude range of the fitting; The best-fit a = 1.25 Schechter function slightly
under-predicts the number of bright galaxies while over-predicting the number of
faint ones.
The completeness-corrected LF, constructed using differential background subtraction and a 0.5 h - 1' Mpc radius aperture to separate the "cluster" and "background" galaxies, is plotted as the open points. The differential background-subtracted
LF is re-normalized to predict the same total number of MR < -17.41 galaxies. The
agreement, on the whole, is excellent. To determine if any variation of the LF with
distance from the cluster center might be causing systematic errors in my results, I
construct a third LF with a differential background subtraction using galaxies having
a projected distance of at least 0.75 h-
1
kpc from the cluster center. Of course, this
biases the sample to galaxies in the more distant clusters where the linear field of
view is larger, and as a result the completeness limit is brighter. The errors in this
smaller sample are also correspondingly larger and I bin the data in 1-magnitude
bins. This LF and the best-fit LF found from absolute background subtraction of the
entire sample are compared in Fig. 11. As a final check between the two different
methods, I fit a Schechter function plus absolute background model to these 969 outer
galaxies brighter than MR = -17.41.
a = -1.00+
This yields yields Mý = -21.03 ± 0.29 and
0.23. The LF of the total catalog and the outer subset are statistically
equivalent. Thus I conclude that, barring an extremely insidious conspiracy between
a sytematic error in the background subtraction and radial variation in the LF, the
two LF constructions arrive at approximately the same LF.
2.4.3
VARIATION OF THE CLUSTER LUMINOSITY FUNCTION
I wish to determine whether there is any significant of variation of the luminosity
function with redshift and the various cluster parameters described in §2.3.2, or if
the cluster LF is indeed consistent with being "universal". The errors in the best-fit
Schechter parameters become progresssively (and depressingly) larger as one sub-
divides the sample down to the level of individual clusters. For this reason, I first
attempt to assess the reality of an effect by simply dividing the 22 clusters into
two bins by the relevant parameter and determining whether there is a statistically
significant difference in the best-fit Schechter values.
I perform two fits for each
sample; one with both Schechter parameters "free", and the second with a fixed to
5/4 to obtain a better-defined M* with which to compare different samples. Errors
are computed only for the latter. The results are tabulated in Table 5.
There appears to be a variation in M*(a = 5/4) such that it is brighter
in the higher redshift sample, brighter in the high density sample, and fainter in
the clusters with Rood-Sastry cD types. A comparison between the 2-parameter
fits of the low- and high-redshift samples and those for a fixed a = 5/4 suggests
that the latter is somewhat exaggerated from using too steep a value of a. Dwarf
galaxies may be disappearing preferentially from the high-redshift sample due to
surface-brightness selection effects, giving a lower observed value. Nevertheless the
data supports luminosity evolution in M* of at least - 0.2 magnitudes, a value
commensurate with the predictions of the passive evolution of early-type galaxies
(Buzzoni 1995). This effect may also explain some of the variation with overdensity
i because of the bias towards richer clusters with higher redshift clusters in the Abell
catalog. In addition, the values of S and M* are covariant in that fluctuations of the
number of the few brightest galaxies in the cluster will affect S and M* in the same
way. The values of M* do not appear to vary with the morphological parameters of the
cluster with the exception of the anomolously faint values of M* for the Rood-Sastry
cD clusters. Note, however that there are only three cD clusters in the sample.
Finally, maximum-likelihood fits of the Schechter LF-plus-background model
were made to each of the 22 individual photometric cluster galaxy catalogs. Plots of
the LFs are included in Appendix B. Inspection of these plots suggests that, with a
few exceptions, this choice of LF performs an adequate task of describing the bright
end of the cluster galaxy LF.
A plot of Mý vs. a (Fig. 12) reveals the strong covariance between the two
parameters. Thus much of the deviation is consistent with an invariant, but noisy LF.
Again, to compare with previous work, I fix the slope a = 1.25 and find the best-fit
value of M,. The distribution of values is shown as a histogram at the bottom of Fig.
12. I also plot these versus the redshift in Fig. 12. There is a strong clustering of
values around
MR
= -21.7, but also a number of fainter values. It seems clear that
the Monte-Carlo simulations are somehow over-estimating the measurement errors by
a factor of - 2.
The values of M* are plotted versus Bautz-Morgan type, Rood-Sastry type,
and the two indices derived in §2.3.2 in Fig. 13. The plots are arranged such that
older, more dynamically relaxed clusters would be expected to fall to the left. The
M* values around -21.7 show a significant trend such that they become brighter to
the left. The values are also correlated with the luminosity overdensity (Fig. 14),
such that M* becomes brighter with hL. This is as would be expected since Ic and
hL are not independent.
2.4.4
DiscussIoN
The agreement between the composite LFs constructed using the absolute and differential background techniques is an excellent indication that there are no serious
systematic errors in the shape of the LF induced by the background modeling in the
former, or a variation in the cluster LF in the latter. The LF in its general form is
well described by the classic, nearly-flat Schechter function as has been found for the
field. There appears to be a significant "dip" in the LF at MRN - 19.5 + 5 log h. This
appears to be general feature of the cluster LF. For example, Biviano et. al. (1995))
find a similar dip in the Coma cluster at Mb < -17.3 + 5log h. Otherwise, the LF
is fairly flat at least to
MR
< -16.5, in good agreement with field surveys and some
work on the cores of clusters.
The LF derived here is inconsistent with claims of a bright upturn in in the
LF at MRN - 17.5 reported in some clusters. At magnitudes much fainter than this
the LF depends critically on the completeness correction which, as I have discussed,
is difficult to quantify. I thus conclude that there is no strong evidence from this
data for a steep faint end to the LF or a dwarf excess. From the fact that there are
22 randomly chosen clusters in the sample I also conclude that any excess seen in
particular clusters is not a general phenomenon. Unfortunately, there is no overlap in
my sample and those in which excesses have been seen, so that it cannot be determined
if the disagreement arises from a difference in the image reduction and analysis.
Assuming a R-r correction of 0.09 magnitudes, the M* derived for this sample
of cluster galaxies with MR < -17.41
is 0.93 + 0.20 magnitudes brighter than the
value derived from the galaxies in the LCRS survey (Lin et al. 1996) (refer to Fig. 1).
However, part of the difference may be due to the detailed fitting of the LCRS LF
and the anomolously flat faint-end slope. The difference between this result and other
field surveys is about - 0.5 magnitudes. That there exists a difference should not
be terribly surprising, given that the morphological distribution of cluster galaxies
are very different from the field. With a fixed to 1.25, the value Mý = -21.47 is
signficantly brighter than nearly all previous values derived from clusters. This exact
amount of discrepancy is not known since the F-R colors of these objects have not
been independently measured. By dividing the clusters into bins with redshift and
morphology, I have demonstrated that there is a tendency for M* to brighten by
0.2-0.4 magnitudes with increasing redshift and denser clusters with a more relaxed
morphology. With the small size of the photometric sample and the well-known bias
towards rich systems at high redshift do not make it possible to distinguish between
these two affects at this time. The evolution is, however, consistent with passive
stellar evolution predictions (Buzzoni 1995).
2.5
BRIGHTEST CLUSTER GALAXIES
The brightest galaxy or brightest two galaxies were identified in each cluster and
were analyzed independently of the image analysis used to obtain galaxy photometry
and described above. Identification of BCGs is a process fraught with peril. The
number counts of foreground field galaxies (see §2.2) are such that on average one
galaxy brighter than R = 15.90 should appear in a field of 75 square arc-minutes
(typical of the survey). This is consistent with the apparent magnitude of the BCGs
in clusters at redshifts from 0.15 to 0.2. Fortunately, true BCGs are located in the
denser centers of the clusters and are fairly homogeneous in their surface brightness
morphology, i.e., are elliptical galaxies with extended profiles. Their inner isophotes
are almost always very round but the outer isophotes may be considerable flatter,
with ellipticities approaching 0.5. Foreground galaxies are usually spirals and will be
located randomly in the field. I found that with a little experience, these two distinguishing characteristics can be used to discriminate against contaminants. There
are also clusters that contain no obvious BCG in which case the brightest galaxy
identified by FOCAS was used.
In a number of cases the BCG candidate fell very close to the edge of the field
of view, several arc-minutes from the cluster center given in the Abell catalog. This
may be due to errors in the coordinates (Huchra, private communication) as well as
physical offsets between the location of the BCG and the projected overdensity of
the cluster on the scale of the Abell radius. Whatever the reason, it implies that a
certain fraction of the clusters in this sample may have BCGs that fall outside the
CCD imaging field. Thus the object that is identified as the brightest galaxy likely
to be a cluster member may not actually be the true BCG.
FOCAS does not lend itself to the analysis of very extended galaxies such as
BCGs. Instead, I use a standard elliptical isophote fitting routine to measure the total
counts to levels below the lo pixel-to-pixel deviations and estimate a total luminosity.
I used the isophote fitting task ELLIPSE in the Space Telescope Science Data Analysis
Softward (STSDAS) package. This routine is based on the isophote analysis technique
developed by Jedrzejewski (1987) in which elliptical annuli of successively larger semimajor axes are fit to the surface brightness profile of a galaxy. The first through fourth
harmonics of the surface brightness distribution within an annulus are calculated. The
centroid, ellipticity, and position angle of the annulus are re-calculated using the first
and second moments, while the third and fourth moments account for any "boxiness"
of the isophotes. I then used the spline-fitting routine in BMODEL to construct a
noise-free image of the galaxy using the isophote data created by ELLIPSE. Very
often the isophote-fitting routine performs poorly in the central few kpc region of
the BCG, especially if there are multiple nuclei or very "boxy" isophotes. This inner
region is usually uncontaminated by other objects and so the data image is used
to replace the model image there. The resulting hybrid image is then subtracted
from the original image to allow FOCAS to detect galaxies close to the BCG and
prevent FOCAS from finding spurious objects in the extended halo of the BCG. The
total counts from the galaxy image is measured and a magnitude calculated. This
is different than the approach followed by Schneider et. al. (1983a) and Postman
& Lauer (1995) and allows us to by-pass the correction factors that are used with a
fixed metric aperture, at the cost of possibly larger random photometric errors and
systematic errors from contaminating objects.
Fig. 15 plots the absolute magnitude M1 of the brightest galaxies identified
as BCG candidates (solid points) from the 22 clusters imaged with photometric conditions. Several features are immediately obvious from this plot: First, there is a
significant locus of the most luminous objects at MRn - 23.6. The outlier with the
extraordinary luminosity is the BCG in Abell 2244, which has a bright foreground
star that is contaminating the isophote fitting and may be responsible for the offset.
Second, the locus appears to have a significant trend with redshift such that more
distant galaxies are more luminous. Third, there is the suggestion of a second locus
at R = -23, albeit with a much larger scatter. Finally, there are a large number of
objects consistent the brightest objects drawn from a Schechter LF.
It is possible that the second "locus" arises from some as yet-unascertained
systematic error in the photometry. It is extremely unlikely that clouds could have
produced such a homogeneous offset over several nights; the effect is too large for
simple haze or variations in atmospheric extinction. The effect shows up in data
obtained from both cameras and is therefore unlikely to have been the product of
a amplifier gain error. I have checked several cases to eliminate the possibility that
poor background subtraction is responsible for the difference. If the second-ranked
galaxies are plotted as well (open points) they are found to have a range of absolute
magnitudes up to the fainter "locus", suggesting that the brightest galaxies falling
on the fainter locus are actually the second-ranked galaxies in the cluster. Either
the true BCGs lie outside the area of the cluster imaged in the survey (as was found
several times to be the case), or somehow a true BCG was prevented from forming.
What remains well defined is a gap of about 0.45 magnitudes between the BCG and
the next-brightest galaxy.
Because of the small size of the photometric sample, and the fact that there
appears to be only a secular trend of luminosity with redshift, I do not try to search
for any correlation of M1 with cluster properties. Instead, I examine the distribution
of cluster properties for those with BCGs and those (apparently) without. In Fig.
16 I plot the distributions with Bautz-Morgan and Rood-Sastry types, and with the
concentration and relative peak indices. Because of the small number of clusters in
the photometric sample it is difficult to distinguish the two distributions, although
the BCG sample does tend towards earlier (more evolved) types. There appears
to be a tendency to find BCGs in lower-numbered Bautz-Morgan type clusters, per
definition. This may either mean the appearence of a BCG in a cluster is unrelated
to its morphology, or, less likely, that there are BCGs in most of these clusters, but
that a large fraction of them are falling outside the field of view of the CCD imaging.
2.5.1
THE M1 - M* DIFFERENTIAL
As discussed in Ch. I, the magnitude difference between the brightest cluster galaxy
and the characteristic galaxy magnitude M1 - M* is a photometry-independent,
distance-independent (except for small differences in K-correction between the BCG
and bright galaxies) quantity that is also of great significance to galaxy evolution
models. As an interesting aside, in Fig. 17 I plot M1 - M2 vs. M1 - M* (M* determined with a fixed a) for all clusters. These quantities are independent since here the
fitting for M* excludes the brightest two galaxies. The clusters occupy a locus which
is well-defined at the bright M 1 end, but poorly defined at the other. I estimate the
values expected from a Schechter LF with the normalization varying over the range
of reasonable cluster richness (10 and 1000 galaxies brighter than M* + 3). The solid
curve is the region that include 90% of the Monte Carlo points. The Schechter LF
occupies a distinct and separate part of this space and the simulations demonstrate
that nearly all of the Mi1 , M2 pairs here cannot be drawn from a Schechter LF, i.e.
that the brightest galaxies in the cluster are not drawn from such a distribution.
Plots of M1 - M* (Fig.
18) demonstrate a weak correlation between this
parameter and the cluster morphology in that more relaxed clusters tend to have
larger differences that asymptote to a value that I define as a BCG. This is clearly
consistent with a picture of a BCG growing with time in a cluster, e.g., through
cannibalism of other galaxies. The scatter is large; a larger photometric sample to
suppress the cluster-to-cluster error in the estimation of M* is obviously needed.
1
Since M* brightens at most only slightly that the value M is responsible for most of
the variation, as is demonstrated in Fig. 19.
2.5.2
DiscussIoN
The first-ranked cluster galaxies, and possibly the second-ranked, are shown to be
distinct from a population drawn from a Schechter distribution. It appears that
many of the CCD fields exclude the BCG (due to offsets between the cluster centers
in the Abell catalog and the BCG locations).
The first-ranked galaxies are - 0.5
magnitudes brighter than all other galaxies and can more or less by identified by
that characteristic.
They exhibit a tight correlation between absolute luminosity
and redshift, showing a brightening of about 0.6 magnitudes by z = 0.35.
The
extrapolated z = 0 value of M1 is -23.25, consistent with estimates by Postman &
Lauer (1995). The very small scatter in BCG magnitudes (if the secular luminosity
evolution can be removed) reinforces their value as "standard candles" for distance
measurement. The appearence of a BCG is correlated with the morphology of the
cluster, suggesting that these objects have evolved (perhaps by cannibalism) over the
age of a cluster but asympotote to some luminosity after some period of time and
only passively evolve further.
I compare the evolution of the ACG absolute magnitude with the models of
Buzzoni (1995). The predicted luminosity evolution is sensitive to the assumed shape
of the Initial Mass Function (IMF) and insensitive to the metalliticty thus a sample
of passively evolving limit objects (such as these might be) can be used to constrain
the IMF in elliptical galaxies. Buzzoni's models assume a single starburst at z = 00
with a power-law IMF of slope -s.
The absolute magnitude evolution with elapsed
time is given by the empirical relation,
dM
d=
C - 0.27(s - 1),
(2.22)
[also (Tinsley & Gunn 1976)]. The elapsed time in an Q < 1 and A = 0 universe is
t=H[- Q L-cosh-'
o°2(0o -1)3/2
- Q +1)+2 -OT z+1 (2.23)
9 0 Z +0 o
0 (1
+ z)
I plot the Buzzoni predictions for three different values of s (1.35, 2.35, 3.35) as
the dashed lines in Fig. 15.
The data would appear to be more consistent with
steep evolution in the model with a very flat IMF, i.e., dominated by massive stars.
It is difficult to draw any strong conclusions from this since there are relative few
BCGs in the sample, the range of redshifts is small, and no complementary color
information is available. Note that choosing a larger value of qO makes objects appear
closer, decreasing their luminosity, but also makes the elapsed time less, lowering the
predicted luminosity of the models. An alternative would be a younger age for the
BCG stellar populations, perhaps produced during the merger and cannibalism of
other bright galaxies.
Finally, I fit a simple form of evolution to obtain a best-fit and remove the
redshift dependence.
Assuming exponential decay of luminosity in an Einstein-de
Sitter universe, and excluding A2244, I find the best fit of the remaining 10 BCGs to
be
Mi(z)-
23.25 - 5.0 [1 -(1 + z)2/3
(2.24)
the residual scatter is 0.06 magnitudes about a mean of -23.31, or 1.1 x 101 1 L®.
This is indistinguishable from the expected random error due to errors in surface
brightness profile fitting (0.04 magnitudes), galactic extinction (0.02 magnitudes),
and atmospheric extinction (0.01 magnitudes). Using the same luminosity evolution,
I find the absolute magnitude of the second-ranked galaxies to be 0.47 magnitudes
fainter.
2.6
2.6.1
MORPHOLOGY OF CLUSTER GALAXIES
THE TECHNIQUE
My objective is to characterize cluster galaxy populations as a whole, rather than
attempt to accurately classify individual galaxies. If accurate, quantitative measurements of the former can be obtained, a comparison with indicators of cluster age or
dynamical state would be a test of the hypothesis that galaxies evolve in clusters
over time, going from a disk-dominated population to a spheroidal population. I
build on previous work (Okamura et al. 1984; Doi et al. 1993; Abraham et al. 1994)
and exploit the fact that the surface-brightness profiles of disk-like galaxies are more
extended and flattened (when not seen exactly face-on) compared to spheroidal systems. I calculate a photometry- and scale-independent quantity that is the difference
of the principle moments of inertia normalized by the isophotal area. In terms of the
moments of inertia M of the surface brightness in image coordinates;
Q
/(Mx
-
Myy)2 + 4M 2
A
(2.25)
where A is the isophotal area. A low value of this parameter indicates the object is
concentrated and/or round, i.e., an early type galaxy, while a large value indicates a
disk galaxy. Since this quantity only measures the surface brightness morphology and
not other qualities such as color, SO-like galaxies will be classified as disk systems.
Fig. 20 is a "Hubble Atlas" of bright galaxies in Abell 2266 at z = 0.1671.
Two-thirds of the clusters in our survey are closer than this object, although the
"seeing" while these images were obtained was better than average (- 0.85 arcseconds). The galaxies are arranged in order of decreasing Q from top left to bottom
right. The top eight objects have the largest Q values in the image, and the bottom
eight have the smallest. Disks systems, especially those seen edge are, and systems
with spiral arms are clearly distinguished from spheroidal systems. The two objects
which appear to be "out of sequence" are (second row, left) a galaxy whose isophotes
are contaminated by a larger galaxy, and (bottom row, second from left) a barred
spiral which is nearly face on and has a high central surface brightness, possible
indicating nuclear activity.
The two most significant systematic errors are the finite "seeing" or telescope
resolution when the images are acquired, and the surface brightness of the detection
isophotes. The redshift of the cluster will enter into both effects; the linear resolution becomes poorer with increasing redshift and the rest-frame threshhold surface
brightness becomes brighter. The net effect of variation in resolution is not intuitively
obvious. Poorer seeing will reduce the quadrapole moments of flattened disk systems
by making them appear rounder. On the other hand, the centrally concentrated profiles of spheroidal systems will become more diffuse and their quadrapole moments
may increase as long as they are slightly flattened. When estimating the preponderance of disk versus spheroid systems in individual clusters, it is clearly desirable to use
a statistic that is robust to modest variations in these seeing and surface-brightness
limits.
To simulate the effects of seeing I convolved the Abell 2266 images obtained
in 0.7-1" seeing with a a = 1" Gaussian filter to bring the effective resolution to
1.4". It was necessary to add noise to the images to restore the background noise
to the original level. Galaxy catalogs are generated from the degraded images and
matched with the original catalogs. Fig. 21 shows a subsection of an image near
the cluster core demonstrating the effects of the poorer resolution. The quadrapole
moments from the original and degraded galaxy catalogs are compared in Fig. 22.
The results are encouraging; even with significant degradation the Q-values for most
objects with logioQ greater than -1.5, i.e., the most extended and flattened objects,
are remarkably preserved. To simulate the effects of a brighter rest-frame surface
brightness threshhold I simply add sufficient noise to the original images to raise the
detection threshhold by a complete magnitude. A comparison of the brightest objects
shows that there appears to be a small effect in the direction of increasing Q, but
that it only a few percent. Finally, I add insult to injury by reducing the surface
brightness sensitivity in the smoothed image by a magnitude as well.
The highest Q objects will be the disk systems observed at oblique angles
or nearly edge-on. These are the "tip of the iceberg" since there will be other disk
galaxies inclined at less favorable angles which reduce their quadrapole moment. I
consider a simple model consisting of randomly oriented disks with identical exponential profiles . There are several competing effects as the inclination of a disk is
varied; the disk becomes flatter, increasing Q, but the surface brightness increases,
moving the limiting isophote outwards and decreasing Q. For a disk inclined at an
angle 0 (where 0 = 0 is face-on),
sin 2 0 [6 - exp(-X) (6 + X + 3X 2 + X3 )]
Q
X 2 [1 - exp(-X) (1 + X)]
where X, the scaled radius of the limiting isophote obeys,
X = -Xo In cos 0
and Xo is the limiting isophote in the face-on case.
(2.27)
The distribution with Q is
plotted in Fig. 21 and is sharply peaked at Q-0.25. High values of Q (order unity
and greater) would not be observed due to the effects of seeing and the disks having
finite thickness.
The paucity of model disk systems with Q < 0.03 and the apparent robustness
of measured Q values above this threshhold suggest that this would be a useful criterion for estimating the relative number of disk galaxies versus spheroids in clusters. I
define the quantity FQ as the fraction of galaxies more luminous than MR = -17.41
and with log
Q>
-1.5 and calculate this using the catalogs from each cluster. Note
that FQ is not truely the relative fraction of disk galaxies since there will be systematic scatter of disk systems out of the cutoff but it is encouring that the numbers
derived are representative of previous estimates of the spiral/SO population in clusters
(Whitmore et al. 1993).
2.6.2
APPLICATION TO THE DATA
Five clusters (A1661, A1990, A2142, A2213, and A2235) were observed twice and
the data reduced independently. A value of FQ was obtained from each catalog and
the RMS variation between measurements of the same cluster was found to be 0.029.
The median number of bright galaxies per cluster is 110, and FQ is typically - 0.3,
and thus the Poisson noise from the finite number of disk-like galaxies is typically
0.05 and the random error dominates any systematic error. (Some of the systematic
variation is probably due to different sizes of the field of view.)
In Fig. 23 FQ is plotted with redshift. A X2 test of all clusters gives a mean of
FQ = 0.280±0.005 and a reduced X2 of 2.39. The scatter is thus marginally consistent
with the estimated errors. To test for any secular variation with redshift I divide the
objects into two bins of equal number (at z = 0.138) and apply the KolmogorovSmirnoff test. I find that there is 6.9% probability that the two subsamples are
drawn from the same population so that the results are inconclusive. As a final test,
I bin the clusters into four redshift bins of 21, 18, 15, and 13 clusters (to partially
compensate for the larger errors at low redshift) and compute the average FQ in
each bin. These are plotted as the solid, connected points along with the data (the
redshift of the bin was chosen to be the median value of the constituent clusters). A
systematic trend with redshift appears, such that the higher redshift clusters tend to
be more disk rich. Without additional information (such as colors) it is not possible to
rule out a systematic error or selection effect. For example, the Abell catalog is very
biased towards richer clusters with increasing redshift and if there was a correlation
with richness and galaxy type (I show below there is no evidence for this) this would
manifest as such a trend.
It is tempting, however, to associate the trend with the well-known ButcherOemler effect, where cluster populations become bluer at redshifts z > 0.2. The
difference between the lowest- and highest-redshift bins is 0.25 ± 0.07 of the average.
In an Einstein-de Sitter cosmology, the elapsed time between the median redshifts
of the bins is 0.88 h - 1 Gyr. If the observed trend were due to disk galaxies being
destroyed or transformed into early-type galaxies, a half-life of 2.1 ± 0.8h-
1
Gyr is
implied.
Figs. 24 and 25 are plots of FQ versus the various morphological parameters
described in §2.3.2.
There appears to be no correlations with an of these cluster
indicators.
In summary, the use of the difference of the principle moments Q of the surface
is a potentially powerful technique to distinguish between disk and bulge systems to
moderate redslhift (- 0.2) and it is robust to mild changes in seeing or noise level. A
simple model of randomly oriented disks suggests a cut-off value to separate the two
classes of systems. Although this technique has not yet been calibrated with images
of galaxies of known morphological types, an application to the data here finds that
the average fraction of disk-like systems is 30%, close to typical values reported in
the literature (Whitmore et al. 1993). A search for a variation in the fraction with
redshift finds a marginal detection of an increase in disk systems at z > 0.2, consistent
with the expected Butcher-Oemler effect (Butcher & Oemler 1984). The utility of
the technique could be greatly enhanced by combing it with color information to
distinguish between SO disk systems and late-type spirals.
0 1
•;0.8
n9.6
E
aX1.4
n
0
0.1
0.2
0.3
0.4
0.5
Fig. 1.- Survey redshifts and filters
Left: Relative imaging efficiency (galaxies observed to a fixed rest-frame surface
brightnes limit per unit time) versus redshift for elliptical galaxies in clusters with
0- 1 projected profiles. The distibution of survey clusters with redshift is shown;
Right: transmittance plots of the two filters used in the survey.
E
c
-1
Fig. 2.- CPE rejection criteria and magnitude errors
Left: Minimum allowable charged particle event flux ratio rejection threshhold
versus FWHM of the image point-spread function before object photometry is
compromised. Right: Galaxy photometry errors from 322 galaxies that were
multiply imaged and analyzed in the field of Abell 1632. The curves are the adopted
la errors.
0Q _)
4
3
o
1
18
20
22
24
26
Fig. 3.- Field galaxy counts
R1,-band galaxy counts from six 100-square arc-minute fields. The solid error bars
are the Poisson errors in the mean; the dashed error bars are the RMS field-to-field
scatter. The solid line is the maximum-likelihood fit over the range R, < 19 < 24.
Number counts obtained by other researchers are shown for comparison. The line
for the bright Weir counts is very difficult to see because of the good agreement!
·I
1.2
01
P.6
0C.9
V
c,
Cf-
a0.2
E
o
U
0.7
0
0.1
0.2
A,
0.3
0.4
18
0.5
20
22
24
Fig. 4.- Effects of Galactic extinction
Left: Total 17 < R < 24 counts versus Galactic extinction AB before any
corrections are made. The line is the best-fit value log N = C - 0.47AB. Right:
Relative completeness of AB = 0.41 field with respect to an average of 5 fields with
< AB >= 0.01, after the magnitudes have been corrected for extinction.
'
1.2
i ii
Y1
r
~i
C\2
V
OJ.9
r--
0
oD
II '
06.5
1
1.5
2
R, [Seeing]
2.5
3
0
'18
20
22
Fig. 5.- Effect of seeing and detection filter width
Left: The effect of seeing (as measured by the radial moment Ir of the point-spread
function) on the normalization. Right: The effect of varying the width of the
detection filter (inversely with the redshift given here) on the number counts of
detected galaxies;
coQ)
Q)
Q)
4
0)
U'
0,
0,
R
Y7
Fig. 6.- Average star counts and background model parameter covariance
Left: Comparison of galaxy number counts with predicted [model of Bahcall &
Soneira (Bahcall & Soneira 1981)], and actual star counts (stars), show that at
R > 21 the star counts are much smaller than the galaxy counts. Right:
Normalization of the field galaxy count model at R = 21 versus the maximum
likelihood faint-end slope 72 over the range 21 < R < 24 for the eight 100 square-arc
minute background fields. The error bars are the 68% confidence limits from Monte
Carlo simulations of the data. The solid point is the maximum-likelihood parameter
pair obtained from the cumulative data, and the locus is the predicted variation in
the case of constant 21 < R < 24 counts.
23
3
2
2
0.1
0.2
z
0.3
1.6
1.8
2
CA
2.2
2.4
Fig. 7.- Cluster luminosity overdensity
Left: Luminosity overdensity JL within 0.5 h - 1 Mpc of the cluster center versus
redshift. Solid points are photometric data. Right: Comparison of JL with Abell
galaxy counts CA within 1.5 h - 1 Mpc. The line is the predicted relation if the
clusters have r - 2 profiles and Schechter LFs. Error bars have been omitted for
clarity.
-
4
1
-
0
23
0~ S
0.5
U.
-
0·
*i·· **
0
*g 0
S
2
0*
·0
**
0
**
· ** ·
-
0
O0
0.1
0.2
z
0.3
0.4
Fig. 8.- Cluster luminosity overdensity and peak index
Left: Luminosity overdensity SL within 0.5 h- 1 Mpc of the cluster center versus
logarithm of the Abell counts. Points are encoded as follows; solid points are
photometric data; decreasing in size are Bautz-Morgan types I to III; triangles,
squares, pentagons, and circles are Rood-Sastry typese I/F, C/L, B, and cD,
respectively. The line is the predicted relation if the clusters have r -2 profiles and
Schechter LFs. Right: Peak index Ip showing the expected bias with redshift.
I I I I I I
I
I
I
I
I
I
I
I
I
I
dyn
S
)
L
O
AzN
A
0
i
A
A
-1
AO
-2
I
I
I
I
I
I
I
I
I
I
I
ill
I
0.5
IP
Fig. 9.- Concentration index vs. peak index
Concentration index Ic versus peak index P1 defined in §2.3.2. Points are encoded
as follows; solid points are photometric data; decreasing in size are Bautz-Morgan
types I to III; triangles, squares, pentagons, and circles are Rood-Sastry typese I/F,
C/L, B, and cD), respectively. The arrow indicates the expected evolutionary track
of clusters towards a more concentrated, smooth morphology.
3
-0-
0
2
1
-22
-20
-18
MR -
5
-16
loglo h
-14
-12
Fig. 10.- Composite luminosity function
Absolute background-corrected LF (solid point) and differential
background-corrected LF (open points) from photometric galaxy catalogs of 22
Abell clusters. The best-fit Schechter function (Mý = -21.13, a = 1.09) is plotted.
The dashed line is the background contribution on the same scale.
sr~e~m~------·r
A
3
•2
.............
...........
..
.....
.....
.. - i......
.... . .................
..
..
..
o.
:
. ,....
,?-......j..--.........i.
.
•.....................
.:.,"
•.,...
_ ...
...
...
...
0
'.-.•.,.'.:'·
,. .: ••- . •.
o
'
....
..
..
..
....
' .€r~,, . ,i;•
_.
.•
,r
• .,...
-21.4
-21.2
21
-20.8
1
-20.6
22
M.
-20
-18
MR - 5 log 10 h
16
-14
Fig. 11.- Error covariance and LF from outer galaxies
Left: Schechter function parameters from 1000 Monte Carlo simulations of the
cumulative photometric data. The circled point is the actual derived value and the
dotted lines are the lor error-bars. Right: Comparison between best-fit Schechter
function using the entire data (solid line) with the LF constructed by differencing
the galaxies inside and outside 0.75 h - 1 Mpc from the cluster center. The dashed
line is the background level.
0.3
is;e--
0.2
0----I
is'-t
0.1
I--~-
0
L
-23
M; + 5 log h
-22
-21
M' + 5 log h (a= 1.25)
Fig. 12.- Schechter LF parameter for clusters
Left: Best-fit Schechter parameters for 22 Abell clusters with photometric data.
The circled X is the point from the cumulative LF. Right: Value of M4 versus
redshift for a fixed a = 1.25. A histogram of M* value is also shown.
83
-22.5
-22.5
B
cD
-22
C/L
I/F
-22
S
**
0
Lo
-21.5
-21.5
-
-I-
*~
SS
-21
-21
*
*
-20.5
................
0
-22.5
-20.5
1
2
3
Bautz-Morgan Type
I
I
I
i
i
I
I
I
I
I
I
Rood-Sastry Type
-22.5
i
I
-22
-22
S*
I~
·
S
-21.5
-
-21.5
*
}OO@.*
.
-f*.
-21
-21
*
II
-20.5
I I
1
I
0.5
I
'
0
IC
'
'
'
-20.5
-0.5
1
0.8
I I
IS
,
I
0.6
I
0.4
*
**S
.
*
0.2
Ip
Fig. 13.- M* dependence on cluster morphology
Characteristic absolute magnitude M* versus Bautz-Morgan type, Rood-Sastry
type, and the concentration and relative peak indices derived in the thesis. The
plots are arranged such that old more dynamically relaxed clusters would tend to
fall towards the left. Only a single, randomly chosen error bar is shown for clarity.
84
nP
I
I
I
C\2
0
LCD
I
C\2
i
I
I I I
2
I I
I
LCD
I ,I I
3
4
1
log 6L (0.5 h- Mpc)
Fig. 14.- M* dependence on cluster overdensity
Characteristic absolute magnitude Mý versus luminosity overdensity with the
central 0.5 h - 1 of the cluster.
5
-25
s
-24
3.35
1.35----- ---
---
--------
s
=
1.3 5
O
m~
-23
E
m
-22
I
_91
0
I
I
I
I
0.1
I SII
I I I I II
0.2
z
II
I I I I I I II II
0.3
0.4
Fig. 15.- BCG magnitudes
Total absolute magnitudes of BCG candidates from 24 Abell cluster galaxies from
24 Abell clusters with photometry plotted against redshift. Filled points are the
brightest galaxies in the field of view that do not have disk-like morphologies. Open
points are the second brightest. The estimated error of 0.05 magnitudes is shown.
The dashed lines are evolutionary models (Buzzoni 1995) with different IMF slopes.
1_
__
10
12
-
10
I
III
II
8
8
6
6
4
4
/
2
i;
2
u
--
10
10
8
8
6
6
4
4
2
2
0O
0
,
0.2
0.4
IC
0.6
0.8
U
0
0.2
0.6
0.4
0.8
1
IP
Fig. 16.- Distibution of BCG and non-BCG clusters
Distribution of Abell clusters with (shaded) and without (unshaded)
photometrically identified BCGs with Bautz-Morgan type, Rood-Sastry type,
concentration index (increasing towards right) and relative peak index (decreasing
towards the right, see §2.3.2 for definition).
I
0
I-1
-2
Q
-6
-4
M - M*
-2
0
Fig. 17.- M 1 - M* vs. M1 - M2
Plot of relative magnitudes of first- and second-ranked galaxies and the
characteristic magnitude M*. The region marked LF is the 90% confidence limits on
simulated Schechter-function clusters with a reasonable range of richness.
_I_ r
............
cD
-3
C/L
I/F
-
S
0
O
0
3
LO-2
+
B
-
a
.
U
-1
-
0
A
liii
-
11111
III
1
2
Bautz-Morgan Type
0
'
-4
'
..
'
'
~ '
'
'
'
'
'
'
.. .. I .
'...
-3
Rood-Sastry Type
`-'"'
S
.
I
I
I
I
I
I
I
I
I
F
S0
0
0*
0
0
0
OS
-2
.0
0
0
*
*·
-1
. .* ..
U U
1
0.5
. .
I. . .
0
IC
I
-0.5
0.8
0.6
0.4
0.2
I'
Fig. 18.- M1 - M* vs. cluster morphology indices
Correlation of M1 - M* with parameters describing the morphology of the cluster
(see §2.3.2). The plots are arranged so that more dynamically relaxed cluster would
appear on the left.
89
-25
II
I
I
I
cD
II
I
B
C/L
I/F
II
" -24
on
*
S
+
*
*
LB
S
**
IS
S
*
6
S-23
-22
2
1
3
4
Rood-Sastry Type
Bautz-Morgan Type
-25
' II
I
-24
'II
I
"
'
· ·
I
I
·
5
SS
I
I
I
I
S
S
*
·
S
I
I
*
S·
0
*
*
I
·
*
-23
-S
*0
-22
Fig. 19.-
I
1
I
I
l
0.5
l
0
Ic
l
i
s t
II
I
-0.5
1
0.8
ll l
0.6
I
lI l
0.4
IP
I l l
Il
0.2
M1 vs. cluster morphologies
Correlation of M1 with parameters describing the morphology of the cluster (see
§2.3.2). The plots are arranged so that more dynamically relaxed cluster would
appear on the left.
90
Fig. 20.- Hubble Atlas of galaxy morphologies by Q parameter
"Hubble Atlas" of 16 MR < -17.41 galaxies in Abell 2266 (z = 0.1671) observed in
0.8" seeing. The galaxies are ordered with decreasing Q, moving from the 8 most
disk-like in the upper left-hand corner to the 8 most star-like in the lower right-hand
corner. Each image is 16.8" on a side. Some of the most compact objects may, in
fact, be stars.
91
&ii
a
aa
44
4: ::_:;is:
Fig. 21.- Image degradation for test of Q parameter
A 96 x 96 arc-second section of an image of the core of Abell 2266 at z = 0.1671.
Clockwise from upper left; original image obtained in 0.8" seeing; degrade to an
effective seeing of 1.4'; poor seeing and a 1 magnitude brighter surface brightness
limit; brighter suface brightness limit only.
-1
-..
Cy
12
,:
0
S
I
*
-3
-
3
--2
-1
logo0Q
0
1
-
*0
4
Ii
I ,
3
I
.
2
log10 Q
-1
...
"
-1
K..
0
S
0 -1
0
-
I
S
0
S
S·
-3
5
-3
3
-2
loglo Q
-2
loglo Q
Fig. 22.- Test of Q parameter with image degradation
The three lower right panels compare the values of Q from the corresponding
degrade images of A2266 shown in Fig. 21 with the values obtained in the original
image. Only galaxies brighter than MR = -17.85 + 5 log h are included. The upper
left pannel is a plot of the distribution of "toy" disk galaxies with Q. The dashed
lines are the adopted cutoff at log Q = -1.5.
93
(\(2
U.U
I I
i
I
I
I
I
I
I
I
I
r
r
0.4
TO3
I
I
0.2
0
\0
I
I I
I
I
0.1
I
I
I
0.2
z
I
0.3
I
I
0.4
Fig. 23.- Fraction of disk galaxies vs. redshift
Fraction of galaxies with logo0 Q > -1.5 (disk-like morphologies) in 67 Abell
clusters versus redshift. Errors bars are from Poisson statistics. The clusters are
averaged into four bins with redshift (heavy error bars connect by the solid line).
0.45
0.35
,
I
.
.
. II
I
I
,
cD
0.4
B
C+L
I+F
0.3
0.35
-
{-
0f
0.3
0.25
0.25
-
-i
S
-
0.2
0.2
Rood-Sastry Type
Bautz-Morgan Type
Fig. 24.- Fraction of disk galaxies vs. cluster morphology
Fraction of high Q (disk-like) galaxies versus Bautz-Morgan and Rood-Sastry
cluster classifications. The values are the weighted averages of the individual
clusters. The trend is from dynamically relaxed, older clusters on the left to
unrelaxed, young clusters on the right.
IS
0.4
So
-
S
,*
6
~
0.
S
•
0
n"
0
*
,
.
..
~~
n~
n i n i
3
2
~0 O0, 0
,
.1
1.5
gO
*
0.2
,,,I,,,I,,,,,,
.••
3.5
I.*
.
·
n
~~
*
'
0.2
0.4
0.6
0.8
Relative Peak Index
Fig. 25.- Fraction of disk galaxies vs. morphology indices
Fraction of high Q (disk-like) galaxies versus concentration and relative peak indices.
Individual error bars have been omitted for clarity; Median errors are shown.
Table 1: Abell Cluster Galaxy Survey
Abella
1
7
24
31
41
84
96
136
175
403
439
1235
1278
1495
1504
1632
1661
1667
1674
1675
1679
1793
1831
1878
1889
1929
1930
1934
1952
1954
1958
1961
1979
1984
z
0.1249
0.1073
0.1338
0.1596
0.2750
0.1939
0.1344
0.1569
0.1569
0.1033
0.1063
0.1042
0.1290
0.1429
0.1836
0.1620
0.1950
0.1648
0.1060
0.1840
0.1699
0.0849
0.0613
0.2540
0.1860
0.2191
0.1313
0.2195
0.2480
0.1810
0.2284
0.2320
0.1687
0.1231
camera
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
Loral
Loral
Loral
STIS
Loral
STIS
Loral
Loral
Loral
Loral
Loral
Loral
Loral
STIS
STIS
Loral
STIS
STIS
STIS
STIS
STIS
filter
RA
RA
RA
RA
RA
RA
RA
RA
RA
RA
RA
RA
RA
Rs
Rs
RH
RA
Rs
RA
Rs
Rs
Rs
Rs
Rs
Rs
Rs
RA
RA
Rs
RA
RA
RA
RA
RA
Sfb
67.1
67.1
67.1
67.1
67.1
67.1
67.1
67.1
67.1
67.1
67.1
268.4
268.4
88.1
88.1
88.1
67.1
88.1
268.4
88.1
88.1
88.1
88.1
88.1
88.1
88.1
67.1
67.1
88.1
67.1
67.1
67.1
67.1
67.1
t
int
1800
1800
1800
1800
1800
1800
1800
1800
1800
900
1800
960
900
600
600
600
2100
600
960
600
600
600
600
600
600
600
1200
1200
600
1200
1200
1200
1200
1200
iPOc
25.2
25.5
25.9
25.1
24.9
25.7
25.4
25.3
25.4
24.7
24.1
24.3
24.7
24.5
24.3
24.4
25.1
24.4
24.4
24.4
24.2
24.9
25.0
24.0
24.4
24.3
24.6
24.2
24.1
24.3
24.2
24.7
24.4
24.8
Abella
1984
1990
2001
2005
2021
*2056
*2061
2065
2069
*2083
*2084
2089
2100
2110
2111
2142
2172
*2175
2178
2183
*2192
*2198
*2211
*2213
2218
2224
*2235
2240
*2244
*2257
*2266
2443
2471
z
0.1231
0.1269
0.1750
0.1275
0.0994
0.0763
0.0768
0.0721
0.1160
0.1143
0.3420
0.0743
0.1533
0.0978
0.2290
0.0899
0.1387
0.0978
0.1429
0.1365
0.1868
0.1696
0.1355
0.1597
0.1950
0.1504
0.1511
0.1380
0.0970
0.1054
0.1671
0.1030
0.1078
camera
STIS
STIS
STIS
STIS
STIS
Loral
Loral
STIS
STIS
STIS
STIS
STIS
STIS
STIS
STIS
S/L
STIS
Loral
Loral
STIS
Loral
Loral
Loral
S/L
STIS
STIS
S/L
STIS
Loral
Loral
S/L
STIS
STIS
filter
RA
RA
RA
RA
RA
Rs
Rs
RA
RA
RA
RA
RA
RA
RA
RA
RA/R s
RA
Rs
Rs
RA
Rs
Rs
Rs
RA/R s
RA
RA
RA/Rs
RA
Rs
Rs
RA/Rs
RA
RA
67.1
268.4
268.4
67.1
268.4
88.1
88.1
67.1
67.1
67.1
67.1
67.1
268.4
67.1
67.1
67.1
67.1
88.1
88.1
67.1
88.1
88.1
88.1
67.1
67.1
268.4
67.1
268.4
88.1
88.1
67.1
67.1
67.1
tint
1200
1270
1050
1200
900
600
600
1200
1200
1200
1200
1200
900
1200
1200
1800
1200
600
600
1200
600
600
600
1500
900
900
1500
900
600
600
1500
1800
1800
Ac
24.8
24.8
24.2
24.9
24.5
25.0
24.7
24.8
24.5
24.6
23.0
24.5
25.1
25.1
24.6
25.3
24.8
25.1
25.0
24.8
24.4
24.6
24.7
25.0
24.0
25.0
24.8
24.9
24.8
24.8
25.1
25.7
25.6
Note. - (a)*indicates observed under photometric conditions; (b) solid angle imaged in arcminutes 2 ; (c) rest-frame surface brightness of 2.5o- detection isophote in magnitudes per arc-second .
Indices
Table 2: Cluster Morphology
__
·
0s
SL
z CA B.M. R.S.
Abell I
II
III
I
2.253
1.160
0001 0.12
51
I
55 II-III
2.593 0.790
0007 0.11
I
III
0.152
3.501
0024 0.13 127
F
0.410
2.702
90 II-III
0031 0.16
L
II-III
0.179
3.272
0041 0.28 153
L
II
2.474 0.731
76
0084 0.10
I
0.000 -0.608
61 I-II
0096 0.13
I
cD
0.806
2.811
99
0136 0.16
III
I
3.268 0.343
84
0175 0.13
1.646
0403 0.10 100 II-III cD 2.116
N.A.
L
-0.417
35
0.000
0439 0.11
II
cD
3.395
0.486
1235 0.10 122
F
3.366 0.149
1278 0.13 151 N.A.
III
3.395 0.178
C
39
1495 0.14
B
3.600
0.218
98 I-II
1504 0.18
I
3.629
0.288
80 II-III
1632 0.16
F
III
0.930
2.291
97
1661 0.19
III
F
3.030
0.198
97
1661 0.19
III
F
0.203
3.070
98
1667 0.16
F
2.729
0.370
1674 0.11 165 II-III
L
3.766 0.238
50 II-III
1675 0.18
3.132 0.455
C
1677 0.18 112 II-III
III
C
2.939 0.302
1679 0.17 115
III
I
3.185 0.229
0.08
54
1793
III
F
3.851 0.231
1831 0.06
67
III
F
3.385 0.325
56
1878 0.25
III
L
3.534 0.128
112
1889 0.19
I
3.557 0.257
95 II-III
1929 0.22
II
0.251
3.714
C
60
1930 0.13
II
0.161
F
3.307
142
1934 0.22
III
0.166
C
3.242
1952 0.25 107
I
0.365
cD
3.468
1954 0.18 120
III
-0.203
I
0.000
88
1958 0.23
III
C
3.224 0.323
1961 0.23 137
II
F
3.123 0.235
1979 0.17 108
II
B
3.748 0.167
96
1984 0.12
··
Ic
-0.723
-0.427
-0.002
-0.602
-0.339
-0.732
-3.079
-0.548
0.004
-1.249
-2.759
-0.085
-0.237
0.574
0.247
0.393
-1.056
-0.317
-0.283
-0.925
0.801
-0.298
-0.507
0.176
0.717
0.355
0.104
0.222
0.644
-0.261
-0.162
-0.002
-3.291
-0.323
-0.286
0.407
IP
0.264
0.793
0.005
0.241
0.154
0.063
0.538
0.854
0.369
0.476
0.757
0.951
0.143
0.008
0.077
0.372
0.095
0.201
0.074
0.231
0.666
0.348
0.452
0.274
0.757
0.242
0.164
0.365
0.210
0.544
0.312
0.866
0.017
0.006
0.662
0.633
RP
0.338
0.346
0.325
0.308
0.328
0.318
0.674
0.305
0.243
0.289
0.259
0.620
0.639
0.347
0.431
0.770
0.929
0.559
0.548
0.279
0.206
0.726
0.204
0.355
0.279
0.776
0.473
0.546
0.488
0.276
0.308
0.582
0.600
0.614
0.390
0.495
Note. - (a)*indicates observed under photometric conditions; (b) solid angle imaged in arc2
minutes2 ; (c) rest-frame surface brightness of 2.5o detection isophote in magnitudes per arc-second
isophote.
Table 3:
Abell
1990
1990
2001
2005
2021
*2056
*2061
2065
2069
*2083
*2084
2089
2100
2110
2111
2142
2142
2172
*2175
2178
2183
*2192
*2198
*2211
*2213
*2213
2218
2224
2235
*2235
2240
*2244
*2257
2266
*2266
2443
2471
Cluster Morphology Indices (cont.)
SL
oa
z CA B.M. R.S.
L
III
3.068 0.183
0.13 140
L
III
3.286 0.205
0.13 140
L
II
3.253 0.204
0.17
57
B
3.562 0.172
III
0.13 105
I
III
52
0.000 -2.307
0.10
50 II-III
C
2.703 0.548
0.08
L
III
4.003 0.200
71
0.08
C
II
3.892 0.190
0.07 109
C
3.952 0.157
0.12
97 II-III
III
cD
3.702 0.307
0.11
60
III
C
3.764 0.176
67
0.34
II
cD
0.07
70
3.792 0.349
III
3.102 0.147
C
0.15 138
cD
I-II
2.926 0.491
54
0.10
C
0.23 148 II-III
2.694 0.274
II
3.446 0.224
B
0.09
89
II
B
89
0.09
3.472 0.219
L
69 II-III
0.14
0.000 -0.261
II
cD
61
0.223
0.10
3.967
I
II
51
0.09
2.551
0.516
II
C
56
0.14
3.701 0.254
62 II-III
F
0.19
3.830 0.201
F
III
85
0.17
2.862 0.187
I
54 II-III
3.082 0.390
0.14
I
III
0.16
2.708 0.308
75
I
III
75
0.16
2.906 0.238
II
C
0.17 214
4.036
0.144
C
III
0.15 138
3.122 0.196
III
F
0.15
73
2.919
0.330
F
III
0.15
73
3.188
0.333
I
III
0.14 165
2.426 0.677
I-II
cD
0.10
89
3.488 0.268
61 N.A.
0.11
C
3.370 0.222
III
0.17
C
83
3.433 0.139
III
0.17
C
83
3.585 0.120
II
0.10 117
C
2.699 0.374
III
0.11
I
2.456 0.902
92
Ip
Rp
-0.491 0.013
-0.273 0.236
0.213 0.211
0.169 0.374
-2.987 0.143
-0.262 0.185
0.836 0.061
0.477 0.294
0.605 0.652
0.632 0.456
0.630 0.230
0.633 0.604
-0.449 0.128
-0.083 0.465
-0.897 0.266
0.148 0.631
0.174 0.024
-3.151 0.178
0.888 0.668
-0.425 0.530
0.671 0.662
0.741 0.437
-0.409 0.322
0.073 0.432
-0.491 0.215
-0.293 0.510
0.232 0.680
-0.429 0.423
-0.264 0.199
0.005 0.446
-1.228 0.073
0.190 0.818
0.291 0.285
0.176 0.038
0.328 0.374
-0.757 0.654
-0.861 0.434
0.252
0.487
0.756
0.526
0.426
0.360
0.297
0.234
0.452
0.251
0.268
0.233
0.519
0.362
0.511
0.242
0.403
0.427
0.387
0.273
0.793
0.282
0.515
0.210
0.423
0.374
0.254
0.716
0.583
0.225
0.572
0.423
0.204
0.350
0.344
0.381
0.296
Ic
Note. - (a)*indicates observed under photometric conditions; (b) solid angle imaged in arcminutes 2 ; (c) rest-frame surface brightness of 2.5cr detection isophote in magnitudes per arc-second 2
isophote.
Table 4: Background Fields
Field Q arc-min.2 a Seeingb
2.08
94.05
0421+019
1.46
92.38
0747+611
1.11
98.87
0953+549
1.23
92.38
0957+557
1.19
102.00
1213-002
1.03
101.74
1245+345
1.45
105.27
1329+412
1.50
98.51
1623+268
1.99
14.61
0747+710
mo
AB [mag]c
0.46 33.60
0.17 32.63
0.00 32.89
0.00 32.82
0.03 32.90
0.02 32.91
0.00 32.95
0.08 32.58
0.14 32.84
E(17 - 24) e
6.47 + 0.26
9.61 ± 0.31
9.32 ± 0.31
10.32 + 0.33
11.09 ± 0.33
11.32 ± 0.33
12.99 ± 0.35
12.21 + 0.35
9.79 ± 0.82
Note. - (a) the solid angle of the field in square arc-minutes after areas around bright stars and
image defects are excised; (b) intensity-weighted radial moment of the image point-spread function;
(c) source: NASA Extragalactic Database; (d) photometric magnitude of 1 DN above atmosphere;
(e) total surface density of galaxies with 17 < R < 24.
Table 5: Variation of LF with Redshift & Cluster Properties
Type
All
z < 0.163
z > 0.163
S<
S>
I/F
C/L
cD
I-II
III
Ic <
Ic >
Ip <
Ip >
M* - 5log h
a
-21.13
-20.85
-21.20
-20.71
-21.25
-21.04
-21.31
-20.14
-21.08
-21.17
-21.10
-21.16
-21.16
-21.04
1.09
1.06
0.91
1.01
0.90
1.00
0.92
0.64
1.05
1.10
1.10
1.09
0.88
1.07
M*(a = 5/4) 11
-21.47
-21.24
-21.94
-21.16
-22.04
-21.55
-22.09
-21.11
-21.50
-21.49
-21.47
-21.51
-21.98
-21.41
+ 0.14
+ 0.10
+ 0.14
± 0.10
± 0.16
± 0.09
+ 0.16
± 0.21
+ 0.11
± 0.10
± 0.09
± 0.10
+ 0.13
± 0.12
Chapter 3
Galaxy Clustering: The Cosmic
Rosetta Stone?
3.1
BACKGROUND
Although galaxies are the stuff of the Universe that is directly observable with our telescopes,
the past fifty-years has seen a steady accumulation of indirect but practically irrefutable
evidence that much or most of the mass is in a 'dark' form that neither emits or absorbs
much light, and may be non-baryonic altogether. Dark matter exists over a range of mass
scales; reviews of the subject include Trimble (1987). The earliest evidence was dynamical
in nature: Dark matter on the scale of galaxy clusters was first pointed out by Zwicky
(1933) and much later on the scale of galaxies (Freeman 1970; Ostriker et al. 1974; Rubin
et al. 1978) based on the dynamics (velocity dispersions) and kinematics (rotation curves)
of galaxies.
Evidence for dark matter on very large scales comes from the analysis of
the deviation of galaxy motions from a uniform Hubble flow ("peculiar velocities") (Dekel
et al. 1993). The amount of dark matter is often estimated in terms of the required mass
density to close the universe. The dark matter in our Galaxy and other galaxies amounts
to Q - 0.03, in clusters is - 0.2 and the large-scale peculiar velocity studies indicate
Q - 0.5. Theoretical prejudices motivate Q = 1 to be consistent with a flat universe with no
cosmological constant (Guth & Pi 1982). By contrast, the total amount of luminous matter
-1
(assuming the mass-to-light ratio of normal stellar populations) contributes QL = 0.0035h
(Lin et al. 1996) and nucleosynthesis theory combined with standard cosmological models
100
limits the total baryonic matter to 0.01-0.05 h - 2 (Trimble 1987).
Galaxies are thus the visible, but perhaps biased, tracers of the underlying and otherwise invisible mass distribution of the Cosmos. One of the goals of modern extragalactic
astronomy is to measure the statistics of that distribution and its evolution with redshift.
Models of cosmological structure formation make predictions of the evolution of this distribution from a postulated field of random mass density perturbations in an otherwise
extremely homogeneous early universe. These models typically consist of three elements:
The world-model parameters Q0 and A0 (the cosmological constant) at the present epoch;
the power spectrum of the initial density perturbations, usually described as a power-law
with index n (which is almost always negative); and a thermodynamic description of the
dark matter, e.g. dissipationless and "hot" or "cold" depending on the temperature at the
epoch when the dark matter particles decoupled from the radiation field.
A detailed review of large-scale structure and its application in testing cosmological
models is beyond the scope of this Chapter. However, a fundamental prediction of all models
in which structure formation is hierarchical, i.e., more massive objects form later than less
massive ones, is that the evolution with redshift is quite sensitive to the value of Qo0. In
"open" models where Q0 is significantly less than unity structure formation is terminated
at z , •o - 1 while for Q = 1 (flat Einstein-de-Sitter model) structure formation continues
to the present epoch. The disparity in the predicted evolution is most pronounced for the
most massive gravitationally bound objects seen at the present epoch. These correspond
to clusters of galaxies with dynamical masses of order 101 5M®.
The different evolutionary histories will manifest themselves in at least two ways:
The co-moving number density of clusters (defined in some systematic fashion) will remain
relatively constant with redshift in an Q0 < 1 universe until z =
-` 1
- 1 after which it will
exhibit marked negative evolution. In an Q0 •1 model the evolution occurs at the present
epoch. Second, the relative dynamical ages of clusters will be markedly different in the two
cases; the to <: 1 clusters will be older than clusters in a flat universe, which are essentially
still forming. In principle, the morphologies of galaxy clusters could be used to statistically
estimate their ages and constrain the value of Qo (Richstone et al. 1992). This possibility
has been a powerful motivation for much of modern research and survey work on cluster
galaxies.
101
3.2
GALAXY CLUSTER SURVEYS
Abell (Abell 1958) developed the first systematic large-area galaxy cluster catalog that,
with its Southern hemisphere and supplements (Abell et al. 1989), contains 4076 objects
identified by eye as areas of high galaxy surface density on photographic plates. The corresponding data in the third dimension (redshift) is sparse; most of the published redshifts of
Abell/ACO clusters are in Struble & Rood (1991). Despite this, the catalog has been the
workhorse of galaxy cluster research ever since. The morphologies of clusters in the original
Abell catalog were characterized by Rood & Sastry (1971), while Bautz & Morgan measured
the relative prominence of the brightest galaxies in these objects (Bautz & Morgan 1970).
Zwicky also constructed a cluster catalog from the same photographic data, but in a
much less systematic fashion (Zwicky 1958). Several other cluster catalogs, selected by silicon rather than by eye, have subsequently appeared (Shectman 1985; Lumsden et al. 1992;
Dalton et al. 1994), all based on relatively shallow photographic plate data. The catalogs
are typically useful in identifying clusters at a statistically meaningful rate to z-0.1 - 0.2.
Deep photographic imaging has been to use to construct small and incomplete samples of
clusters to z < 1 (Gunn et al. 1986; Hoessel & Schneider 1985). These deeper catalogs
suggest that the comoving number density of galaxy clusters evolves very little to z - 0.5.
Spectroscopic and CCD imaging surveys of galaxies provide a far superior quality of
information, but have been limited to much smaller areas of the sky. However, as the size
of these samples have steadily increased there have been reports of significant clustering at
redshifts of 1-3 (Le Fevre et al. 1994; Giavalisco et al. 1994). To develop an intermediate
catalog of reasonble depth and size, Postman et. al. (1996) completed a CCD survey
of 5 square degrees in two optical pass-bands. Although their catalog contains only 107
cluster candidates without redshift information, they find that their data is consistent with
a constant co-moving number density of clusters to z , 0.6 that is
-
5 times higher than
the density of rich Abell clusters at the current epoch.
The identification of a cluster as a gravitational bound object can be secured by
spectroscopic measurement of the recessional velocities of candidate member galaxies. In
the absence of such detailed (and costly) information, the reliablity of a galaxy cluster catalog, and any conclusions drawn from it concerning numbers of clusters and their evolution,
depends on the statistical quality of the catalog, and the degree to which the objects iden-
102
tified in the catalog correspond to physical clusters. These problems can be distilled down
to three basic issues:
The contamination of the catalog by objects which are spurious enhancements
of the projected surface density of galaxies caused by the chance superposition of many
smaller overdensities, or the projection of single large but gravitationally unbound overdensities along the line of sight, e.g., the sheets seen in large redshift surveys (De Lapparent
et al. 1991).
The completeness of the catalog, i.e., the ratio of the number of clusters detected
to the actually number at a given detection limit. There are any number of mechanisms
through which incompleteness could arise: Juxtaposition of clusters with bright stars may
prevent an accurate count of the galaxies or their accurate photometry and eliminate such
objects from the catalog. A superposition of a cluster with a significant underdense region
("void") along the line of sight will suppress the galaxy counts and may cause the cluster to
drop out of the catalog altogether. If the shapes of galaxy clusters depart significantly from
spherical symmetry (Plionis et al. 1991), projection effects become important. For example,
if clusters are actually filametary in form, they will be detected most easily end-on when
the projected surface density of galaxies is highest.
The correspondence between the projected clustering of galaxies in space and
the underlying; mass distribution; i.e., do gravitationally bound objects on mass scales of
1014 - 1015M, always have corresponding and proportional enhancements in the galaxy
density?
The concept of galaxy 'bias' was introduced (Bardeen et al. 1986) to explain
why galaxies appear to cluster much more strongly on scales of about 8 h -
1
Mpc than the
underlying mass distribution does, as inferred from the galaxy velocity dispersion. If galaxies
preferentially form in the densest areas the discrepancy is nicely resolved. In the case of
cluster selection, one is not concerned with the relatively small amounts of bias postulated
to explain the large-scale observations. If, for some unknown reason, there was a significant
amount of negative bias for some clusters, e.g. galaxies failed to form or were destroyed in
them, then there might be clusters with very high mass-to-light ratios that would fail to
appear in optically-selected catalogs. These mythical creatures, so-called "dark clusters",
are an intriguing but unsubstantiated idea. A dark cluster might reveal itself through the
gravitational lensing of background objects. Dark lenses have been implicated as the culprit
of the wide-separation QSO lense candidate 2345+007 (Steidel & Sargent 1990). A more
103
secure identification of a dark lense might occur through the detection of X-ray emission
from a hot intracluster medium in hydrostatic equilibrium with a gravitational potential
that contains few or no galaxies (see the next section).
Perhaps the best candidate yet
found is the "dim" cluster HCG94, identified as a compact group by Hickson et. al. (1989),
but found by Ebeling et. al. (1995) to possess an X-ray luminosity commensurate with a
full-fledged cluster of galaxies.
3.3
3.3.1
X-RAY CLUSTERS
X-RAY EMISSION FROM CLUSTERS
X-ray emission from the Coma galaxy cluster (z = 0.023) was first confirmed by a sounding rocket flight in 1969 and reported in 1971 (Meekins et al. 1971).
Uhuru, the first
X-ray observatory satellite, subsequently detected X-rays from several other clusters and
obtained spectra that showed the X-ray spectra to be consistent with free-free (brehmstrahllung) emission from optically thin intracluster medium at temperatures of a few keV
(Kellogg 1973). Successive satellite missions have enormously expanded both the number of
X-ray clusters and the amount of detailed information about their emission. The increased
sensitivity of X-ray instrumentation has enabled the detection of clusters to cosmological
distances: Currently, the most distant X-ray cluster lies at z = 0.81 (Gioia & Luppino 1994).
The Coma cluster has an X-ray luminosity in the 0.5-4.0 keV range of 2.0 x 1044 h - 2 ergs
s - 1, where h is the Hubble Constant in units of 100 km s-1Mpc - 1 . The detection limits of
the latest imaging satellites like Einstein and Rosat with modest exposure times is 1 x 10-13
ergs s - 1 cm - 2 , and thus for an open (qo = 0) Robertson-Walker cosmology, a Coma-like
cluster could be in principle be detected to a redshift of 2.3.
Observations and some simple physics indicates that; (1) the X-ray emitting gas is
in approximate hydrostatic equilibrium with the gravitational well of the cluster; (2) the
emission is long-lived and not a transient phenomenon.
Pressure-supported gas in hydrostatic equilibrium with the gravitational potential
of a cluster will have a temperature approximated by the virial relation;
T2mp a 2 6 .6keV
3kTB = 6.6keV
104
0km/sec
'
(3.1)
where a is the galaxy velocity dispersion in the potential well of the cluster. The close
correspondance between the observed gas temperature and galaxy velocity dispersion supports the hypothesis that the X-ray-emitting gas in most clusters is equilibrated.
This
phenomenon has been explored to to place significant constraints on models of the depth
and shape of the gravitational potential and the mass (much of which is unseen) responsible
for it, e.g. (Buote & Canizares 1994; Buote & Tsai 1996).
The X-ray volume emissivity of gas with primordial composition (76% H and 24%
He) at temperature T > 1 keV is Ex = 4.87 x 10-24nniT1 /2 ergs sec-lcm
-3
K - 1/ 2
(Lang 1980). The total luminosity of a cluster with gas-mass Mg and electron temperature T, obeys the scaling,
Lx ~ MgpTe/2
(3.2)
The density-squared dependence of the emissivity has profound consequences:
First, it
means that the inner regions of the cluster with the highest gas density will be greatly
accentuated, making X-ray clusters appear as distinctive, clearly-defined objects on the sky.
Supposing that the space density of gas has the radial profile r- 2, the corresponding X-ray
surface brightness will fall off as 0-3, much faster than the projected galaxy distribution
(0-1). The corollary of this effect is the small volume filling factor of the X-ray emission
(-
10-6).
The probability of chance superpositions becomes vanishingly small and thus
samples of X-ray selected clusters do not suffer from the projection effects which hinder
optical surveys. On the other hand, the sensitivity to the gas density makes the X-ray
emission very dependent on the thermal state of the gas, as I show later.
The approximate cooling time of the X-ray emitting gas is simply the mean thermal
energy divided by the luminosity:
L
1 Looma
1r
~2Lm,m =8.3Q3kTM
M
1015Me
L
)
(
)
0.03
Gyr
(3.3)
i.e., of order the Hubble time and much longer than the crossing time (- 109 years). Only
in the innermost regions of some clusters is the estimated cooling time shorter than the
dynamical time, producing "cooling flow" instabilities (Fabian et al. 1991).
Since the emission from X-ray clusters is long-lived, can be detected to high redshift, and is not subject to line-of-sight projection effects, the statistics of these objects
are a potentially powerful tool to test models of cosmology and structure formation (Eke
105
et al. 1996).
One is also relying on a phenomenon that is different than the process of
galaxy formation and thus may be a means of detecting dark clusters missed by optical
observations, i.e., the X-ray emission of a cluster is not directly dependent on the cluster
mass-to-light ratio.
The X-ray picture is not all that rosy, however. The very effect which makes X-ray
clusters easy to find, i.e., the sensitivity of the luminosity to the gas density, also hinders an
interpretation of the observations. Since clusters are detected and selected based on X-ray
flux, and thus, at a given redshift, their intrinsic X-ray luminosity, the thermodynamic state
of the gas will significantly bias a catalog (I discuss the effects of X-ray surface brightness
in the next section). This state may depend on the thermal history of the gas, which is very
uncertain. Radiative cooling can be very important in the centers of some clusters at the
present epoch where the density and brehmstrahllung emissivity are high (Fabian et al. 1991)
and this process may have been more important at earlier, times when collapsed objects
would have been denser. High rates of star formation and subsequent supernovae during
the formation of galactic bulge populations may have injected energy into the intracluster
medium at an earlier epoch. Finally, it is not known to what degree the gas is shocked by the
merging of dark-matter halos in the hierarchical picture of structure formation. This may
depend on the power spectrum of density fluctuations, since the merging of large masses
would be expected to produce more shocking of the gas, while the steady of accretion of
small masses might produce a more adiabatic evolution.
I illustrate this by presenting a paradox: The simplest model of X-ray cluster evolution includes the hierarchical formation of structure produced by an initial field of Gaussian
fluctuations with spectral index n evolving in a universe with critical mass density (Q = 1).
Clumping proceeds to larger and larger mass scales with time, and the gas is continually
heated by shocks to achieve hydrostatic equilibrium. The typical mass of a bound object
at redshift z scales as;
M ,
(1 + z) - 6/(n+3).
(3.4)
The collapse density is proportional to the mean cosmological density at that epoch;
p _ (1 + z) 3
106
(3.5)
Combining these with the virial equation gives the temperature of the gas;
T , (1 + z) ( n -
1) / ( + 3)
n
(3.6)
.
Thus the X-ray luminosity of a characteristic bound object is,
L,
MpT 1/ 2
,
(1
6
+ z)(7n+5)/(2n+ ).
(3.7)
These scalings are supported by the results of high-resolution numerical simulations (Navarro
et al. 1996). For the Cold Dark Matter model, the spectral index over the relevant range
of mass scales is n = -1 and therefore Lx,
(1 + z) - 1/2, decreasing with redshift. However
the co-moving number density of such objects would increase rapidly as M
-3
-
(1 + z) 3 .
Thus the co-moving X-ray emissivity of the universe would increase with redshift. In an
open universe ' < 1 the rapid positive evolution would still occur, but not until a redshift
Z
Q- l- 1.
As has been pointed out by Evrard & Henry (1991) and Kaiser (1991), among
others, this is firmly ruled out by the observations, which if anything, suggest negative
evolution at z < 0.5. One explanation is that the spectral index is steeper, e.g., n = -2,
such that L, - (1 + z) - 9/2. However this does not explain the lack of evolution seen at at
optical wavelengths (§3.2). An alternative suggestion (Kaiser 1991) postulates an injection
of energy into the intracluster medium at some early time, raising the entropy of the gas
to a much higher value. This might have occured during the epoch of galaxy formation
at z > 3 (Steidel et al. 1996). Thereafter, the gas evolved adiabatically. Taking the ratio
-5
of specific heats y to be 5/3, the X-ray luminosity is found to evolve as Lx , (1 + z) .
There is an unfortunate degeneracy in a high sensitivity to both the power spectrum and
adiabaticity. Furthermore, the effects of negative evolution in the X-ray gas can be partially
offset by the suppression of structure formation in an open universe.
A different approach is to select clusters based on the temperature of the X-ray gas
(usually done by fitting models to X-ray spectra). The temperature of the X-ray emission
is a direct indication of the depth of the potential well and thus the mass of the cluster.
(But this does not eliminate the luminosity and surface-brightness selection effects in the
sample.) Also, since the X-ray-emitting gas traces the shape of the cluster potential, the
surface brightness morphology, which is independent of the X-ray luminosity, remains an
107
independent test of cosmology.
Several techniques have been applied to measuring the
shapes of nearby clusters (Mohr et al. 1995; Buote & Tsai 1996). Presumably, clusters at
higher redshifts would be younger on average and thus appear more distended or irregular
on the sky.
Exploitation of the detailed properties of high-redshift clusters for the purposes of
constraining cosmological models must await the next generation of X-ray telescope satellites which combine high sensitivity, spatial resolution and energy resolution, particularly
the Advanced X-ray Astronomical Facility (AXAF) (Canizares 1990). In the meantime, it
remains to be clearly discerned what the results of current and previous surveys, however
imperfect, are telling us.
3.3.2
X-RAY CLUSTER SURVEYS
Once it was confirmed that the X-ray emission from most galaxy clusters is the thermal
free-free emission from an optically-thin intracluster medium, the number of detected clusters and our increased understanding of their statistics and evolution with redshift has been
limited only by the steady improvements in sensitivity and resolution brought about by advancements in X-ray telescope and detector technology. This has also brought an increased
understanding of the inherent systematics of the catalogs which are produced (§3.4).
The large numbers of galaxy clusters detected by the X-ray telescope and the Imaging Proportonal Counter (IPC) on the Einstein Observatory satellite made possible the first
serious investigations of their statistical properties (Henry et al. 1982; Soltan & Henry 1983)
(see descripton of Einstein and the IPC in §4.1.1). These found that the X-ray luminosity
function (XLF) of nearby Abell clusters could be described by a power law over the range
1043 - 104 5 ergs sec- 1 , with more luminous clusters having a lower co-moving number density. The value of the power-law index is close to -2, depending on the details of the fitting
process. Soltan & Henry (1983) found that L, correlated with increasing Abell "richness",
a measure of the galaxy density, evidence that the X-ray luminosity was an indication of
the global state of a cluster, and not an independent statistical process. Henry et. al.
(1982) concluded that there was no evidence that the slope of the XLF evolved strongly
with redshift to z - 0.5, although the small size of their sample and incompleteness still
allowed for considerable evolution.
Edge et. al. (1990), using a larger and more complete X-ray cluster sample con-
108
structed by Lahav (Lahav et al. 1989), found that a Schechter function with characteristic
1
luminosity L* = 2.0 ± 1h - 2 x 1044 ergs sec - ('
1Lcoma) provided a slightly improved
fit to the XLF and a slightly flatter slope of a = -1.65 ± 0.26. They also measured the
X-ray temperature function (XTF) using measurements from the EXOSAT observatory and
obtained a best-fit power-law index of a = 4.93 ± 0.37. Finally, the sample was found to be
significantly depleted of Lx > L* clusters at 0.1 < z < 0.2 compared to the space density
of such objects at z < 0.1, suggesting a steepening of the XLF with redshift. No attempt
was made to constrain any redshift evolution of the XTF.
A significant advancement in the size and quality of X-ray surveys occured with the
advent of the Extended Medium Sensitivity Survey (EMSS), which was constructed from
sources serendipitously detected in Einstein Imaging Proportional Counter images of high
Galactic latitude targets (Gioia et al. 1990; Stocke et al. 1991). Of the - 800 objects, 104
were eventually identified as galaxy clusters (Gioia & Luppino 1994). The most distant
cluster in the sample lies at z = 0.81.
1.5 x 10-13 ergs s -
1
cm
-2
A complete subset of these above a flux limit
and a declination S > -400 were anlyzed (Henry et al. 1992):
The clusters were divided into three approximately equally-populated bins with redshift
(0.14 < z < 0.2, 0.2 < z < 0.3, and 0.3 < z < 0.6). The slope of the XLF was found
to monotonically steepen with redshift. A power-law fit to each XLF returned indices of
-2.09 ± 0.20, 2.63 ± 0.22 and 3.09 ± 0.27 in successively higher redshift bins. This provided
strong evidence for negative redshift evolution in the luminosity or number of clusters with
Lx > 0.7 x 1044 ergs at z > 0.2.
In 1991) the Roentgen Observatory Satellite Rosat, was launched. The satellite, still
in limited operation at this writing, carries an X-ray telescope with three detectors; two
Position Sensitive Proportional Counters (PSPC) and a High Resolution Imager (HRI). The
reader is refered to Ebeling (1993) and references therein for a detailed description of the
satellite and instruments. The X-ray telescope/PSPC combination offered an effective area
3 times larger than the Einstein telescope and IPC, with an on-axis angular resolution of
25", a factor of 3.6 improvement. The circular field of view of the PSPC was 2 degrees
in diameter, although the imaging performance of the X-ray telescope deteriorated rapidly
further than 20 arc-minutes from the axis. The PSPC was used to conduct a six-month
All Sky Survey (RASS) in which - 50, 000 sources have been detected. Subsequent pointed
imaging has covered several percent of the sky and resulted in the detection of another
109
70, 000 sources.
The ROSAT Brightest Cluster Sample (BCS) consists of a flux-limited (f,[0.1 2.4keV] > 5.5 x 10 - 12 ergs sec -
1
cm - 2 ) sample of northern (6 > 200), high Galactic
latitude (JbJ > 200) galaxy clusters, most of which were identified by their association with
previously known (i.e., optically identified) clusters. Ebeling et. al. (1996) constructed an
XLF from these clusters and fit a Schechter function, finding L* = 2.2 ± 0.4h sec -
1
2
x 1044 ergs
and a = -1.78±0.09, in agreement with Edge et. al. (1990). However, they find that
their larger sample is consistent with no evolution in the XLF to z < 0.3. It appears that
the Edge et. al. result was probably the result of a statistical "fluke" de-populating the
high-redshift portion of their sample. However, the Ebeling et. al. result is not in confict
with the evolution found in the EMSS at z > 0.3.
In fact, samples of clusters from the deeper RASS field around the North Ecliptic
Pole, which extend to lower flux levels and higher redshifts, do appear to show negative
X-ray luminosity evolution at z > 0.3 (J. P. Henry, private communication).
There has
been a proliferation of projects to identify galaxy clusters in PSPC pointed images, which
have flux limits an order of magnitude deeper than the RASS (Castander et al. 1994; Rosati
et al. 1995). The samples are still small and conclusive results from these efforts are still
awaited.
This is to be contrasted with the picture at optical wavelengths of little or no evolution in the co-moving number density of clusters to much higher redshifts. This raises the
"correspondence" question; are the objects seen in X-rays the same as those in the optical?
At low redshifts there is evidence that this is true. Ebeling (1993) and Ebeling et. al.
(1993) have constructed catalogs of rich Abell clusters detected in the Rosat All Sky Survey
and in the process have found that nearly all rich clusters have bright X-ray counterparts to
z , 0.15. There are a number of serendiptiously observed X-ray cluster, particularly nearby
clusters, that fail to appear in the optically-selected catalogs. At the moment there is no
substantial published evidence that the two populations are significantly different. Tucker
(1995) searched for "failed" clusters of galaxies using 17 unidentified sources in Einstein
IPC images, 10 of which showed significant source extent. Of these ten objects, nine were
found to be associated with "significant" clustering on POSS photographic plates and a
ninth was identified as a star. [This author independently obtained CCD imaging of the
remaining object which showed it to be a distant rich galaxy cluster.] They concluded that
110
failed galaxy clusters do not consistute a significant fraction of the over-all cluster total.
Thus the disparity between the X-ray- and optically-selected samples is either a
real affect, or a redshift-dependent systematic of one (or both) of the catalogs. The pitfalls of optically-selected cluster samples were discussed above; I next discuss the problems
associated with the X-ray surveys.
3.4
LIMITATIONS OF X-RAY SURVEYS
Our view of the X-ray evolution of clusters is affected by the biases and selection effects
inherent in surveys. One such bias is the energy range of the X-ray detection, which will
bias the results towards clusters with characteristic temperatures hotter than the energy
range. The effect is small for the Einstein IPC and ROSAT PSPC telescope/detector combinations with significant sensitivity in the 0.5-4 and 0.2-2 keV ranges, respectively. A more
significant bias which I address here is the surface brightness of the X-ray emission. Clusters with a given total flux which are resolved by imaging detectors can be more difficult to
detect. The improvement in X-ray optics has brought a succession of telescopes with better angular resolution: However this skews detection towards sources which appear equally
more compact on the sky. If clusters evolve rapidly with redshift, then surface brightness
selection effects may be important in X-ray surveys. A sample of X-ray samples will also be
biased by the particular machinery used to detect the objects in X-ray images. Traditional
X-ray source detection techniques have employed a "'sliding box" or window of fixed size
which is moved over the image to find locations of significant excess count rate with respect
to the background. The detection process will be biased towards those objects with surface
brightness profiles similar to that for which the detection window geometry was optimized,
e.g., point sources. The production of the EMSS catalog employed such a point-source
detection algorithm and as a consequence, the sample of galaxy clusters in the EMSS is
biased towards clusters with a high central surface brightness. More extended sources with
integrated fluxes above the survey threshold levels may have been excluded due to their
lack of concentrated emission. Henry et. al. (1992) used isothermal P models with /, the
ratio of gas temperature to velocity disperions equal to two-thirds, and found that between
50% and 93% of the cluster emission falls outside the 2.4' x 2.4' detection cell of the EMSS
survey.
111
Many clusters exhibit a centrally peak X-ray surface brightness distribution which
has been linked to the development of a "cooling flow", a process where the gas in the
dense inner core of a cluster attains a cooling time shorter than its dynamical time and
continuously cools as it moves towards the bottom of the potential well (Fabian et al. 1991).
The peak in the surface brightness may contribute only a small fraction (- 10%) of the
total flux from the cluster, but may increase the central surface brightness by an order of
magnitude. Pesce et. al. (1990) have analyzed this effect and determined that 80% of
the Lahav et. al. (1989) sample of 53 bright X-ray clusters contain cooling flows. They
extrapolate this to the Medium Sensity Survey (MSS) clusters and, noting that the MSS
detection was tuned to objects with peaked surface brightness profiles, conclude that the
MSS sample is reduced by a factor of four if the estimated contributions by the cooling
flows are removed.
Samples of very bright, nearby clusters which are constructed in a manner that is
not biased with surface brightness would be at the very least instructive about what one
might find in fainter, more distant samples. The Rosat Brightest Cluster Sample (BCS)
consists of 172 clusters with f, > 5.5 x 10-12 ergs sec-1 cm - 2 [0.1-2.4 keV] and Ibi > 200
detected in the Rosat All Sky Survey (Ebeling et al. 1996). The original sample of 164
X-ray clusters were detected at the locations of known clusters using the Standard analysis
Software System (SASS), (Voges 1992). However, when the fluxes of these clusters were
calculated using the VTP algorithm (Ebeling 1993) in 2 x 2 degree-fields, an additional 8
clusters were serendipitously discovered. [Ebehling (1994) developed the Veroni Tesselation
Program (VTP) to detect low surface brightness objects of arbitrary shape in ROSAT PSPC
images and has used this to great effect in the X-ray photometry of galaxy clusters. VTP
computes a local density around each detected X-ray photon and links photons in high
density areas together to construct objects.] Multiplying by the ratio of the relative areas
covered (- 6.3), this number suggests there would be a total of about 50 serendiptious
clusters in the survey.
A correction to this estimate must be made to account for the correlation of clusters
of galaxies as projected on the sky; The number of serendipitous detections near other
clusters will be somewhat higher than in random fields. The spatial correlation function of
the BCS is
(),
1 8()13.7h-1Mpc)
.
112
(3.8)
(Bahcall & Cen 1994). The depth of the survey, i.e., the median cluster distance, is approximately 210 h- 1 Mpc; The survey is extemely dilute due to its large depth. The angular
correlation function is then approximately (Peebles 1993),
W (0)
13
.8 1-1.8
]
2\ -10
(3.9)
where 0 is in radians. Thus w reaches unity at a 0 of only 7.4 arc-minutes. I integrate over
a circular field with a 4 square degree area to find an excess probablity of 28% of finding
an X-ray cluster in a field centered on another X-ray cluster with respect to random fields.
Based on this, the expected number of X-ray clusters missed by the SASS anaysis would
be - 39, or 24%. However, the 95% confidence limits on this incompleteness are 11% and
32%.
The archiving of X-ray imaging data from recent and current telescope missions
and the exponentiation of computational power has permitted more careful analyses and
re-analyses that are able to detect X-ray objects in a manner less biased with surface
brightness. Besides Ebeling's VTP algorithm, Rosati et. al. (1995) have used wavelet
analysis to search for extended objects as cluster candidates in deep pointed ROSAT PSPC
images. In the next Chapter I describe a flux-limited catalog of objects constructed from
the entire collection of Einstein IPC images. While the Einstein IPC has been superseded
by the ROSAT PSPC in sensitivity and angular resolution, the systematics of the former
have been better characterized and direct comparisons can be made to previous catalogs
such as the EMSS with no need to correct for changes in the energy range of detection.
The detection machinery used to find objects is less biased in surface brightness and thus
the catalog can. generate a more complete survey of galaxy clusters. In addition, the new
catalog retains some surface brightness profile information about each object. Information
like this may be useful in quantifying the morphologies of X-ray clusters.
113
Chapter 4
An Archival EINSTEIN IPC X-ray
Cluster Survey
The archiving of the entire Einstein IPC database, and the improved understanding of the
telescope and detector systematics has led to re-analyses of the data aimed at pushing the
limits of source detection.
Hamilton & Helfand (1993) performed a fluctuation analysis
of several IPC deep fields to constrain the number counts of sources immediately below
the detection limit of the images. Moran et. al. (1996) have pushed this much further,
constructing a catalog of nearly 5 x 104 20r source candidates in 2520 high Galactic latitude
fields.
Here I describe sources detected in Einstein IPC images in much the same fashion,
except that the objective is to construct a catalog that is optimized for the detection of Xray clusters in that it is much less biased with surface brightness than previous IPC source
catalogs. An improved characterization of the XRT/detector response reduces sysematic
errors and permits the significance of sources to be assessed over larger solid angles. Thus
sources of a given flux but greater angular extent and lower surface brightness are less likely
to be rejected. In §4.1 I describe the construction of the catalog, source identifications,
and follow-up observations in §4.2. I then make several cuts of the catalog in an effort to
identify promising galaxy cluster candidates (§4.3).
114
4.1
4.1.1
THE SOURCE CATALOG
Einstein IPC
The Imaging Proportional Counter (IPC) was one of the primary instruments on the High
Energy Astrophysical Observatory (HEAO)-2 Einstein satellite which operated for two and
a half years from November 1978. The IPC was a gas scintillation imaging detector and the
X-ray telescope/detector combination was sensitive to photons in the energy range 0.5 to
4.0 keV with an effective area of about 100 cm 2 . The field of view was 76 arc-minutes on a
side with an on-axis resolution of 1.5 arc-minutes. During its operation, the IPC obtained
nearly 4100 images with exposure times ranging from 100-56,000 seconds. The celestial
coordinates of every photon detected in IPC images are recorded in optical disk archives at
the Columbia University Center for Astrophysics.
4.1.2
SOURCE DETECTION
The raw source catalog was provided by D. Helfand and B. Oppenheimer (Columbia University). The source detection algorithm was based on previous work (Hamilton et al. 1991)
for analysis of I[PC Deep Survey fields. Details of the algorithm and its application to the
construction of a 2a source candidate catalog is described in Moran et. al. (1996). Briefly, a
cumulative event list and a corresponding exposure map was constructed for each spacecraft
orbit using all I[PC events from the unobstructed 38 x 38 arc-minute center field of view,
with additional restrictions on the energy channels used and the allowable telescope-Sun
angle during the observations. Individual flat fields are constructed for each orbit and account for energy-dependent vignetting and spatial variations in the detector response. The
computed count rates for all orbits are then summed into cumulative count and count-rate
maps.
The maps are binned into 32 or 64 arc-second square pixels and scanned with a
series of four circular apertures with different diameters: 2.5, 4.7, 8.4, and 12.2 arc-minutes.
The smallest aperture is the optimal size for the detection of point sources and the largest
aperture is limited by considerations of the field of view. The background was estimated
in a circular annullus surrounding each aperture; the width of each annulus is proportional
to the diameter of the aperture to maintain a constant ratio (about 14.5:1) of background
115
area to detection area. The background contribution in the aperture is calculated as the
background level multiplied by the ratio of areas and the noise was estimated by adding the
Poisson noise from the counts in the aperture and the expected background contribution
in quadrature.
A detection and its location were recorded if the counts in an aperture
exceeded the background by the noise multiplied by a threshhold factor, and if there was
an acceptable percentage of reliable pixels in the aperture and background (roughly 60%
and 30%, respectively).
Detection proceeds iteratively over the entire sky map, with the
threshhold decreasing from an intially high value. Pixels associated with sources at each
iteration were masked out in successive iterations. The entire map was analyzed separatedly
with each aperture (using 32 arc-second pixels for the smaller two apertures and 64 arcsecond pixels for the larger two).
An initial pass was made through the IPC sky map using a 2.5o detection criterion.
This threshhold is set deliberately lower than the final detection criterion so that more countrate information is available for the different apertures when sources are finally generated.
2.5a was considered the lowest threshhold practical since still lower levels generate too
many spurious detections for the catalog to be generated in a reasonable time (24 hours).
A catalog of sources was constructed from the four separate lists of detections, by matching
them if the center of one detection fell within the aperture of another. Each source then
consisted of the multiple detections. If more than one detection in the same aperture was
matched, the detection with the lower signal-to-noise was discarded.
The final detection
criteria were established by examining the detector coordinates of all sources and raising
the threshhold for detection until locations of high frequency of spurious detection (e.g.,
near the IPC window support ribs) were eliminated. The final significance threshholds are
3a, 4a, 4.5a, and 5o in the 2.5, 4.7, 8.4, and 12.2 arc-minute apertures. The signal-to-noise
must exceed this criterion in at least one aperture for there to be a detection. The increasing
threhhold with aperture size reflects the need to account for larger systematic errors.
If the error in the location of a detection is dominated by Poisson noise, then it is
proportional to - R/a, where R is the aperture radius and a is the signal-to-noise. The
location of each source was defined to be the centroid computed as the weighted average of
all the aperture centers xi;
2
X- =Eii(R/a)Ej(Ri/ori)- 2
116
(4.1)
A total of 7419 sources were identified in this manner. To serve the purposes of
this project, only sources with Galactic latitude Ibi > 150 were retained. Also, many nearby
extragalactic objects have diffuse X-ray emission and produce spurious detections that could
be identified as extended sources. Any sources within 50 of the Large Magellenic Cloud,
2.670 of the Small Magellenic Cloud, 2.670 of the Coma Cluster, or 1.60 of Messier 31 were
excluded. The final IPC source list contains 6610 sources. Fig. 1 is a log N-log S (cumulative
number versus Ilimiting count rate), where the count rates are those from the 8.4 arc-minute
aperture. The number counts of an idealized survey of homogeneously distributed, nonevolving sources to a uniform depth has a number-count slope of -3/2. The departure of
the IPC source counts from this behavior is mostly due to the non-uniform survey depth,
with the solid angle of sky imaged decreasing with count-rate limit.
4.1.3
SOURCE IDENTIFICATION
The IPC source list was correlated with catalogs of known or prospective X-ray sources
to identify known sources such as X-ray emitting stars and Active Galactic Nuclei (AGN),
as well as previously known X-ray clusters.
These are the Einstein Online Catalog
(EINLINE), a, compilation of catalogs of identified X-ray sources detected by the Einstein
Observatory; the NASA Extragalactic Database (NED), an on-line database of extragalactic objects maintained at the Infrared Processing and Analysis Center (IPAC) ; and
SIMBAD an on-line catalog of both Galactic and extragalactic objects maintained at the
Centre des Donnees de Strasbourg (CDS).
In the case of prospective point sources, the matching criterion were established by
the two-point angular correlation function for the X-ray source-candidate pairs alone. Fig.
2 plots the relative offsets between the X-ray source and the nearest EINLINE point-source
match (star, AGN). There is the expected concentration of matches around the origin whose
shape describes the distribution of random and systematic errors in the source position.
The concentration has a circularly symmetric component plus a 'tail' of significant matches
extending over 2 arc-minutes from the origin. This tail may arise from a systematic error in
IPC positions, or from a pecularity in the point response function of the X-ray telescope/IPC
combination. To include these matches, the area within a match is described by a circle of
1.5 arc-minute radius and an abutting square 2 arc-minutes on a side.
A source is considered 'identified' if the match is with an object that is a plausi117
ble X-ray source and unrelated to an X-ray cluster. For example, AGN, very bright stars
(my < 8), emission-line or variable stars, and nearby galaxies fall into this category. Unidentified radio sources, IRAS (Infrared Astronomical Satellite) sources, and distant galaxies
(e.g., Zwicky galaxies) could be associated with galaxy clusters and thus are retained as
candidates.
The X-ray sources were correlated with the Abell (Abell 1958; Abell et al. 1989)
and Zwicky (1958) catalogs of optically identified galaxy clusters. Determining matching
criterion for optically identified clusters is more problematic, since the optical centers of
these objects were determined from the centroids of the galaxy distribution on large scales
(often in a subjective manner), and these may be significantly offset from the centroid of any
X-ray emission. In addition, there may be multiple X-ray detections within the cluster. Fig
2(b) is a plot of the offsets between the IPC sources and Abell clusters with contour plots
overlaid. There is a central concentration within 8 arc-minutes of the Abell coordinates
and an extended "halo" of matches extending to about 18 arc-minutes.
Some of these
may be produced by multiple detections, bimodal clusters, or by the strong cluster-cluster
correlation function. If one relaxes the offset criterion for matching to include these there
is the danger of incurring large numbers of spurious sources as the number density of IPC
sources on the sky is roughly 2 per square degree. Thus I adopt a maximum offset of 8
arc-minutes (heavy line in the figure) for Abell and Zwicky clusters.
4.1.4
SOURCE EXTENT TEST
The unidentified IPC sources are doubtlessly dominated by objects other than X-ray clusters, e.g. AGN and stars. In the absence of detailed X-ray spectra that can discriminate
between the soft- and power-law- spectra of these objects and the
-
6 keV thermal emission
of clusters, the spatial extent of the source can be used to find promising cluster candidates.
X-ray cluster exhibit a wide range of surface-brightness morphologies, but a standard model
of X-ray clusters (Cavaliere & Fusco-Femiano 1976) is one where the mass distribution of
the cluster is described by an isothermal sphere (p - r - 2 ) and the gas is in hydrostatic
equilibrium with the gravitational potential. The X-ray surface brightness is then
E = Eo
1+ (r
1 ro 1231/2
118
(4.2)
where the / is the ratio of the specific energy in the galaxies in the potential to that of
the gas and has a canonical value of 2/3. The core radii ro of nearby clusters is found to
range upwards from a minimum of 50h - 1 kpc, with a typical value of 100h - 1 kpc (Jones
& Forman 1984). This subtends - 45',' one-half of the FWHM of the Einstein IPC, at
z = 0.15.
Thus one might hope to discriminate between point sources and clusters to
approximately this redshift.
In this catalog, the information on the spatial extent of the sources is in the form
of the relative amount of flux in the four apertures. I use this information in a statistical
test that returns the probabiity that a source is extended. However, three cautionary notes
are in order: First, the catalog is dominated by low signal-to-noise objects for which it
may be difficult to robustly separate point-like and extended objects. Second, the width
of the XRT/IPC point response function is energy dependent and decreases with higher
photon energy. Thus point sources with 'softer' spectra and lower-energy emission will have
a more extended appearence in IPC images. Finally, any catalog of X-ray cluster candidates
selected on image extent will systematically exclude high redshift clusters with concentrated
emission since these will have remained unresolved by the IPC.
Following Helfand & Oppenheimer (private communication), source extent is evaluated by comparing the signal-to-noisevalues between apertures rather than the count-rates.
For reasons that are still not clear, the distribution of point sources with signal-to-noise
Rij = oi/aj ratio is more sharply defined than the count-rate ratio for known point-like
objects. This is demonstrated in Fig. 3 and Fig. 4 where the count-rate ratio and signal-tonoise ratio distributions are shown for known point sources (stars and QSOs) with a > 10 in
the 2.5 arc-minute aperture and measurements in the three other apertures. Note there are
only three independent choices among the six possible ratios, however six separate criteria
are necessary since not all aperture information is available for each source. The extent criteria are established by selecting by eye the three ratios which exhibited the most marked
cut-offs in the point-source distribution and calculating the other ratios from these. The
former criteria are R 21 > 1.20, R 32 > 1.14, and R 4 3 > 1.03; the latter are R 31 > 1.37,
R 4 1 > 1.41, and R 4 2 > 1.17. For comparison, the ratios of sources identified with Abell
clusters are shown in Fig. 5. A large fraction (but not all!) of these sources satisfy one or
more of the extendedness criteria. In the limit of perfect measurement, a source which satisfies any of these criteria is considered anomolously extended at the corresponding angular
119
scale. These criteria are optimized to eliminate as many point sources as possible, rather
than to detect as many X-ray clusters as possible. There are simply far too many sources
to examine many more than the most promising candidates.
In the presence of finite noise, the confidence with which a source may be considered
extended depends on the significance with which the criteria are satisfied, i.e., how far an
Rij exceeds the threshhold value in terms of the error in Rij. It is desirable to assign a
quantitative estimate of this confidence to each source, allowing one to make cuts in the
catalog depending on one's particular objective or degree of ambition in performing followup observations. Performing this task analytically is extremely difficult since one has to
account for systematic detection biases (e.g., Malmquist bias) that become important near
the detection limit of the catalog. Instead, I have elected to use Monte Carlo technique
to test the significance of source extent. Each of the sources is simulated 10000 times,
taking the measured count rates to be the "true" count rates and adding random noise.
Each simulated source is passed through the detection criterion and rejected if it fails. The
extended source criterion constructed using the high signal point sources are then applied
to the "detected" sources. The ratio of the number of simulated sources which pass the
extendedness criterion to the total number which are "detected" is an indication of how
significant the source extension is.
In addition to choosing the values of the ratios in signal-to-noise for the extendedness criteria, one must also set the number of aperture pairs for which the criteria must be
satified. The Monte Carlo results were used as a guide for setting this requirement. I found
by experimentation that requiring only one signal-to-noise ratio be significantly large did
not sufficiently discriminate against known point sources. Too few sources met the extendedness criterion for all three independent ratios (many sources are missing a measurement
in an aperture due to an insufficient number of good pixels). Requiring that two ratios be
extended appears to be a happy medium.
Most of the sources are near the detection limit of the Einstein IPC where the noise
will be dominated by the particle background. If this is well characterized, the fractional
error in the ratio of signal-to-noise between two apertures is equal to the fractional error in
the ratio of count rates:
(ci/cj)
a___ -
Rij
= C(4.3)
Ci/Cj
120
The count-rate measurements in each aperture are not independent since, in assembling the
individual detections to construct sources, only heavily overlapping apertures are matched.
In assessing the statistic significance of a ratio of count-rates this dependence must be taken
into account and the count-rate ratio must be calculated in terms of independent quantities.
From a list of count rates and positions for each aperture it is not possible to
reconstruct the true statistics. Here I use a simple model of a source to approximately
account for this effect. I asssume that the apertures are perfectly concentric, such that the
independent measurements are the difference in count-rates between successive apertures,
i.e., the count-rates in concentric annuli. The relations between the aperture count-rates
Ci and annuli count rates cij are:
Ci = C 1 +- E
cij
(4.4)
The associated variances in each quantity are:
rEi- 1
ci
- 0@1
j=
C2
C1 + c12
C,
C1
(4.5)
cY 3
(4.6)
With some math, one can show that:
2
=Rij
= S? + S-
2
1-
C,
(4.7)
23
where Si is the signal-to-noise in the ith aperture. Thus the variance in Rij is smaller than
the independent measurement case, as expected, but approaches the latter for Ci > Cj.
Gaussian-distributed errors are used in the Monte Carlo simulations. Each simulated source is checked for detection as well as spatial extent. I calculate two probability
estimators, one for detection and the second for extent;
PD = ND/NMc
(4.8)
PE = NE/ND
(4.9)
where NMC is the total number of Monte Carlo simulations, NDo is the number of detected
121
simulations, and NE is the number of detected simulations that appear extended. PD and
PE give the probability that a source with those particular statistics would be detected,
and the probability that, if detected it would appear extended.
The cumulative distribution of all objects with PE, the probability of extent, is
shown in Fig. 6. Stars and AGN identified in the EINLINE catalog are plotted separately
to calibrate PE. There is a nearly linear correspondence between the threshhold PE and
the number of stars and AGN below that value, validating our error models and Monte
Carlo simulations. 98.5% of stars and AGN have PE < 0.99. As a population, AGN appear
slightly more extended then stars; a result opposite to the effect expected from the energy
dependence of the XRT/IPC resolution. The cumulative distribution does show, however
that the tail of the PE distribution for stars is more pronounced than for AGN.
Of particular importance in understanding the statistics and limitations of the extendedness test are identifying the tail of the tail of the point-source distribution, i.e., those
known point sources which meet a conservative extendedness criteria. There are seven
sources in the IPC catalog with PE > 0.99 but which lie within 2 arc-minutes of X-ray
sources listed in the EINLINE catalog (Table 1). This is 1.5% of the 476 EINLINE stars
and AGN in the IPC catalog. One object is a white dwarf and two others are main sequence
stars; their high PE can be understood in terms of their very soft spectra: At low energies
the point-response function of the X-ray telescope/IPC combination is broader and sources
can appear extended.
Oddly, this does not appear to be the case with the majority of
stars, as I have shown. An unidentified UV-excess object may also have a soft spectrum
in X-rays. The AGN mistaken for extended sources require a different explanation. Two
of the three have relatively low S/N in their 2.5 arc-minute apertures and may simply be
statistical flukes. Other possibilities include the scattering of X-rays from dust in the interstellar medium to produce an X-ray "halo" (Predehl & Klose 1996), or systematic errors in
sources near the edges of the IPC window.
Finally, it is important to bear in mind that by selecting with PE one is biasing the
sample toward sources with signal-to-noise sufficient to discriminate between the point-like
and non-point-like possibilities, and thus in favor of brighter objects.
122
4.1.5
THE EMSS SURVEY REVISITED
The EMSS catalog was also generated from a survey of X-ray sources in high Galacticlatitude EinsteinIPC images (Gioia et al. 1990), and is nearly completely optically identified
(Stocke et al. 1.991). Of particular importance are the 104 galaxy clusters it contains, 99
of which have spectroscopic redshifts (Gioia & Luppino 1994).
Thus it behooves me to
correlate and compare the IPC sources catalog constructed here with the EMSS.
I matched the IPC source catalog with the EMSS catalog maintained online at the
Centre de Donnees de Strasbourg (CDS). Of the 835 sources in the EMSS, 576 have an
IPC source within 2 arc-minutes, and 76 of the 104 EMSS clusters have counterparts. The
fraction of all EMSS sources not appearing in the IPC source list here versus signal-to-noise
is shown in Fig. 6. This fraction is not a strong function of signal-to-noise. Rather, the
missing sources are probably lost due to the restriction of this survey to the inner 38 x 38
arc. min. field of view of the IPC. The EMSS used the entire field of view minus a central 10
arc-minute diameter aperture centered on the target, and the area shadowed by the support
ribs. The EMSS area per field was 0.6 deg 2 as compared to opposed to 0.4 deg 2 here. The
missing field is is consistent with the fraction of missing sources found here (29%), although
the latter is somewhat lower due to the decreased sensitivity in the excluded regions of the
[PC.
Fig. 7 compares the corrected count-rates in the EMSS with the 2.5 arc-minute
aperture rates; the agreement is very good and the scatter is within the errors (only a
typical pair of error-bars is shown for clarity). The few outlying points, particularly the
clusters, arise from multiple detections within extended sources. The advantage of detecting
objects using the larger apertures, particularly for X-ray clusters, is shown in Fig. 7(b),
where the maximum signal-to-noise among the four apertures is compared to the EMSS
signal-to-noise.
Plotting the probability of extent PE values of the re-detected EMSS clusters versus
their redshifts proves telling (Fig. 8). Only 16% of the cluster have PE > 0.99 and all are at
z < 0.25. Only 26% of the clusters are found extended at two sigma confidence (PE > 0.954)
and still only 43% at one sigma confidence (PE > 0.683). Only one cluster at z > 0.3 is
extended at a the 2a level. These results are consistent with the early estimate based on a
X-ray cluster core radius of 100h - 1 kpc. They also indicate that samples selected with PE
should not expect to see objects similar to EMSS clusters beyond z - 0.3.
123
4.2
4.2.1
OPTICAL COUNTERPARTS
DIGITIZE SKY SURVEY IMAGING
Preliminary searches for galaxy cluster counterparts to some of the Einstein IPC sources
were performed using Digitized Sky Survey (DSS) images. The DSS was created by the
Space Telscope Science Institute by digitizing Schmidt photographic plates using a microdensitometer and compressing the data by a factor of about 10:1 for distribution on
CD-ROMs. The Northern hemisphere plates are Palomar Observatory Sky Survey (POSS)
103a-E (red) 2400-4200 second exposures plates obtained during from 1951 to 1955. The
Southern hemisphere plates are SERC J Southern Sky Survey plates with exposures of
1800-7200 seconds. The limiting magnitudes are (very) approximately R - 20 and Bj - 21.
While the red colors of galaxies, particularly the early-type galaxies in clusters, favor their
detection in the E plates, the J plates are of signficantly better quality.
Fifteen by fifteen arc-minute square DSS images of Abell clusters with richness class
2 at various redshifts are shown in Figs. 9 and 10. (Unfortunately some of the detail in
the images is inevitably lost in the compression and manuscript reproduction). While, all
details about the galaxies are lost at redshifts greater than 0.1, the clusters can still be seen
as a concentration of low surface-brightness features to z = 0.3 in both the E and J plates,
and thus it would appear that one can find any rich cluster counterparts to these sources
at least to this redshift.
4.2.2
CCD
IMAGING OF GALAXY CLUSTER COUNTERPARTS
Optical observations remain the touchstone for the confirmation and characterization of
X-ray clusters. These observations consist both of spectroscopy to verify that there is a
significant excess of galaxies exist within a narrow range of recessional velocities (a - 1000
km sec - 1 ) and imaging with Charge-Coupled Devices (CCDs) or infrared arrays to measure
the numbers, color, and, resolution permitting, morphologies of the cluster galaxies. An
observing program to obtain CCD imaging of selected unidentified, high-galactic latitude
X-ray sources discovered in Einstein IPC images was begun in 1994 using the 1.3 meter
McGraw-Hill telescope at the Michigan-Dartmouth-MIT Observatory and the 60-inch Meyer
Telescope at the Palomar Observatories of the California Institute of Technology.
124
The
sources are selected based on their relatively high significance of detection and favorable
location on the sky, and not necessarily on their extendedness. The primary objective of
the project is to confirm that a significant spatial concentration of galaxies exists at the
location of the X-ray emission.
A secondary goal is to obtain photometry of likely cluster members in two or more
pass-bands and establish their apparent magnitudes and colors. Accumulating evidence
describes the majority of clusters galaxies in terms of two distinctive populations. The first
is a population of early-type galaxies that have undergone little change other than than the
passive evolution of the old, evolved giant stars which produce most of their luminosity and
appear to have originated in a single episode of star formation at large redshift . The objects
in a particular cluster exhibit a remarkable degree of uniformity in color, indicating that
they are coeval (Stanford et al. 1995). Furthermore, there seems to be little or no variation
from cluster to cluster. In a color-magnitude diagram the early-type galaxies in a cluster
(essentially all at the same distance from the observer) lie along a straight line called the
"C-M relationship", with fainter galaxies being slightly bluer. The slope is thought to arise
from a variation of metallicity with luminosity. This relationship defines an edge or "red
envelope" (O'Connell 1987) the red side of which is devoid of cluster galaxies. The apparent
passive evolution of the elliptical galaxies means that the secular evolution of the observerframe colors with redshift is dominated by K-corrections arising from the redshifted galaxy
spectra. In principle, the colors of these galaxies can be used to make photometric estimates
of the redshift of the cluster (Molinari et al. 1990; Molinari et al. 1994).
The second population is composed of blue galaxies with spectra indicating recent
or ongoing star formation and disk-like morphologies ((Butcher & Oemler 1984; Dressler
& Gunn 1992; ]Dressler et al. 1994). These objects have roughly disappeared in clusters at
the present epoch, but show a steadily increasing presence in clusters at z > 0.2, called the
"Butcher-Oemler effect".
Intermediate-band Gunn-Thuan filters were selected for this work since a considerable amount of distant galaxy photometry has been done using the Gunn-Thuan photometry
system, e.g., (Hoessel & Schneider 1985; Molinari et al. 1990). Transmission plots of gri filters obtained from the Kitt Peak online filter database (http://www.noao.edu/kpno/filters/filters.html)
are shown in Fig. 11(a). Colors generated in this photometry system are useful for separating ellliptical and spiral galaxies. Fig. 11(b) is a plot of the predicted g - r color of a
125
passively evolving elliptical galaxy versus redshift taken from Buzzoni (1989). The different
curves respresent different ages of the galaxies at the epoch in which they are observed.
Essentially, the observed colors become redder as the large Balmer break from the evolved
stars moves to longer wavelengths. Initially, Gunn-Thuan g and r filters were used; an i
filter has been added to avoid the potential g - r degeneracy with redshift in the range
0.4 < z < 0.6.
Two observing runs were conducted at MDM in February 1994 and April 1995, and
one run took place at Palomar in December, 1995. The February 1994 imaging was taken
under photometric conditions and I present some results from those observations here. A
Tektronix 20482 CCD with 21 pm pixels was used in 2 x 2 binning mode for a final scale
of 0.635" per pixel. The seeing ranged from 1.3 to over 2 arc-seconds. Exposures were 600
seconds through g and 900 seconds though r.
The images were processed with IRAF (Image Reduction and Analysis Facility)
routines. The images were bias-subtracted using the overscan region and trimmed. Dark
sky flats for each night were constructed by combining all of the images taken in a particular
filter on that night, removing objects and saturated pixels by excluding a number of the
highest and lowest values at each pixel.
Charged particle events were removed by the
COSMIC routine, in which each pixel is compared to its neighbors and replaced by the
average if it exceeds a threshhold of four standard deviations above the background and
50% above the mean of the neighboring pixels.
(These values were found by extensive
experimentation with images).
Short-exposure images of Gunn-Thuan spectro-photometric standards were obtained.
The atmospheric extinction coefficients listed in Kent (1985) were adopted; i.e., ag = 0.179
and ar = 0.0806. The adopted calibration magnitudes for one count per second above
the atmosphere are g = 21.8 and r = 22.55. Galactic extinction in the B-band were obtained from NED and converted to g and r band extinction assuming Ag = 0.9AB and
Ar = 0.7AB. The 2.5a detection isophote in the cluster candidate images is approximately
r = 25.5 magnitudes per arc-sec 2 with the telescope at the zenith.
The Faint Object Cataloging and Analysis System (FOCAS) was used to detect
objects, map limiting isophotes, and calculate various photometric and surface brightness
quantitites.
A detection isophote limit of 2.5a was adopted.
The r images, with their
deeper detection limits, were used to define the isophotes of the objects.
126
The g-band
images were then transformed to the pixel coordinates of the respective red images and the
common isophotes used to evaluate g-magnitudes of objects. This allows for a more rigorous
determination of galaxy colors since the flux is summed from the same solid angle of the sky
in the different pass-bands. Fig. 12(a) compares the g magnitudes of 63 galaxies generated
using the r isophotes against those using independent g isophotes. The deviation from
perfect agreement with fainter magnitude is the flux lost due to the g detection isophotes
shrinking. To determine whether slight offsets of the images or the effect of the finite-sized
pixels might be causing a systematic error in the g magnitudes I compared the g - r colors
determined by common isophotes and those determined through 6.4" apertures. Fig. 12(b)
demonstrates that any systematic error is negligible compared to the scatter.
As example of a typical optical counterpart to an extended IPC source and to
illustrate the potential of combining X-ray and optical observations, I present an analysis
of follow-up imaging to the X-ray source EX0806.3+2057.
This source was detected at
only 3.3a significance in the 2.5 arc-min. aperture but 5.7a in the 12.2 arc-min. aperture
with a count rate of 0.021 sec - '.
It is assigned an extent probability of 93.5%.
The
neutral hydrogen column density in this direction is nH-2.2 x 1020 cm - 2 (Stark et al. 1992).
Assuming a 6 keV Raymond-Smith spectrum the X-ray flux in the 0.3-3.5 keV range is
5.6 ± 1.0 x 10- 13 ergs sec - 1 cm - 2 .
Fig. 13(a) is a 4x4 arc-minute section of the field of the X-ray source EX0806.3+2057
centered on the X-ray emission. North is up and East is to the left. The cluster has two
bright galaxies to the south and north-east of the X-ray centroid, each surrounded by a concentration of fainter objects. There is also a third disk galaxy to the south that is probably
a foreground object.
A color-magnitude diagram of the galaxies within 2 arc-minutes of the X-ray center
is shown in Fig.
13(b).
Only galaxies brighter than r < 22.5 are plotted since this is
roughly the corresponding detection limit of the g image. Several features of the g - r color
distribution of the r < 22 galaxies are evident. First there are a few very blue objects with
g - r < 0.4, including the r = 18.9 galaxy with an obvious disk morphology. These have
colors consistent with those of lower-redshift spiral field galaxies. There is a broad peak in
the distribution centered at g - r - 0.6, and a narrower peak at g - r-1.1 which includes
the brightest galaxy in the 16 arc min 2 field.
The mean color of the five brightest galaxies in the red peak is g - r = 1.10 ± 0.02
127
(assuming that uncorrelated error is responsible for the scatter of 0.05 magnitudes about
this value).
From (Molinari et al. 1990) this suggests a cluster redshift of 0.30 to 0.33
depending on whether the galaxies are as young as 8 Gyr or as old as 12 Gyr at the epoch
they are observed. I compute the fraction of blue galaxies following the classical procedure
(Butcher & Oemler 1984), with some modifications for the different band-passes involved.
I adopt the absolute magnitude limit of My = -20 (h = 0.5) and q - 0 = 0.5. I use the
photometry tranformation in (Molinari et al. 1990); V - r = 0.723(B - V) - 0.468 and
assume B - V = 1.0 for the reddest (E/SO) and B - V = 0.4 for the bluest (spiral Scd and
irregular) galaxies. For z = 0.3 K-corrections I use Kr = 0.40 for E/SO galaxies (Schneider
et al. 1983b) and 0.24 as an estimate for Scd galaxies. When the dust has settled, the
cut-off has taken the form r < 22 - 0.42(g - r - 0.4) and is plotted as the dashed line in
13. The vertical dashed line is the adopted color separation into red (g - r > 1) and blue
(g - r < 1) galaxies. Correcting for the slope of the C-M relationship at this redshift, a total
of 22 and 20 galaxies fall into the respective categories. The background is estimated from
galaxies in the image outside of 5 arc-minutes (1.5 h-
1
Mpc if zc = 0.3) from the X-ray
center. The total background galaxy count to r - 22 is 1.3 per arc-min. 2 , consistent with
other estimates (see Chapter II). Estimated background counts in the inner 4 arc-min are
4.2 ± 1.5 red galaxies and 7.8 ± 2 blue galaxies. Thus the fraction of blue galaxies is - 40%
but with uncertainties of 15% due to the small number of galaxies involved. The positions
of the blue and red galaxies are plotted separately in Fig. 13.
The total absolute magnitude of the brightest red galaxy, assuming z = 0.315,
qo = 0.1 and an r-band K-correction of 0.4 magnitudes (Hoessel & Schneider 1985), is
Mr = -22.37 ± 0.03 + 5 log h.
Postman & Lauer (Postman & Lauer 1995) found the
average absolute magnitudes of Abell cluster BCGs within a 20 h - 1 kpc diameter aperture
to be Mr = -21.27 with a scatter of 0.32 magnitudes.
To compare with this result, I
assume a redshift of 0.315 and compute an angular diameter of 10.6". The corresponding
aperture magnitude is Mr = -21.86 ± 0.03 + 5 log h. Thus only a few tenths of a magnitude
of luminosity evolution is required to make this consistent with the low-redshift results.
However, the g - r color of 1.24 for the most distant BCG in the Hoessel & Schneider
(1985) survey at z = 0.2890 is somewhat redder than what is found here. If one accepts
a photometric redshift of z - 0.3, the X-ray luminosity of the cluster candidate is - 7 x
1043 h - 2 ergs sec - 1 [0.3-3.5 keV], respectable for a galaxy cluster. The 4 arc-minute field
128
represents a linear distance of 1.2 Mpc.
4.3
X-RAY CLUSTER CANDIDATES
4.3.1
BRIGHT SOURCES
As discussed in 3.4, a significant number of serendipitously-identified nearby X-ray clusters
with fluxes above 5.5 x 10- 12 ergs sec-1cm -
2
[0.1-2.4 keV] have been detected by Rosat
which do not appear in optical catalogs (Ebeling et al. 1996). An independent search for
X-ray bright, non-Abell clusters using sources serendiptiously identified in the extended
source analysis of Einstein IPC data would be useful.
The equivalent emitted flux in the Einstein 0.3-3.5 keV band is 6.9 x 10-12 ergs
sec -
1
cm
- 2.
To convert this into an IPC count rate, I assume the canonical cluster has a
6 keV thermal bremstrahhlung spectrum, lies at a redshift of 0.1, and is observed through
a neutral hydrogen column density of 4 x 1020 cm - 2 . I find an IPC count rate of 0.22 cts
sec - '.
Unfortunately, the sky coverage of the AEXCS at this flux limit is roughly 1100
deg 2 (excluding cluster pointings) and thus one is hopelessly dominated by low-number
statistics. The expect number of Abell and non-Abell clusters at the flux limit is - 5 and
3, respectively. In the serendipitous (i.e., non cluster target) IPC pointings, I find 2 Abell
clusters, 1 non Abell cluster, as well as a single unidentified extended source (see below).
To these I add the target clusters, their numbers weighted by the ratio of the serendipitous
sky coverage to the area of the entire sky above Ibi = 150 (a factor of 0.036). There are 47
Abell clusters and 9 non-Abell clusters in the targeted images, leading to equivalent totals
of 3.7 and 1.3. The only firm conclusion that can be drawn from this is that the extended
source analysis of the IPC images does not lead to the discovery of a significant excess of
bright X-ray clusters on the sky over the numbers already detected by previous Einstein
and Rosat analyses.
4.4
SOURCES WITH PE > 0.99
I next examine those unidentified sources which appear significantly extended in IPC images
and are the most promising candidates for z < 0.3 clusters of galaxies. A combined SIMBAD
129
and NED search at the coordinates of the 121 sources in the IPC source list with PE > 0.99
found 14 galaxy clusters, 8 galaxy groups, the nearby galaxy Messier 101, 2 EMMS sources
identified as galaxies, and an additional 35 sources identified with nearly point-like sources
such as AGN and stars, individual galaxies, and ultraviolet sources. A measurement of
the random coincident match rate for these sources was conducted by shifting the input
declinations by 1 degree. I found that with a 3 arc-minute matching radius, a conservative
choice designed to eliminate all possible point sources from the catalog, the rate of spurious
matches is 15%, or 18.
The two EMMS sources, MS1224.7+6733 and MS2357.5-6352, have offsets of 4.3
and 5.6 arc-minutes from the X-ray source centroid. Both are identified as low-redshifts
AGN (z = 0.153 and z = 0.136, respectively). The optical identification of the EMSS sources
appears to be unambiguous (Henry et al. 1992). It is possible that additional background or
foreground sources are mimicing an extended source. The forty-three unidentified extended
sources are listed in Table 2. DSS images of these indicates about 1/3 are possible clusters
within reach of the photographic plates.
130
3
2
O
0
1
n
3
-2
-1
0
log Count Rate
1
Fig. 1.- IPC Source Counts
Log N-log S plot of the high galactic latitude sources detected in IPC fields. The
count rate in the 8.4' aperture is used. The -3/2 slope for a uniform survey of
homogeneously'-distributed, non-evolving sources is shown for comparison.
131
4
¢.
:
•
. :
. . .
D.
:••.
in Minutes of' Arc
Offet
" . '.
":..
•
.-
,
•
-2--4
-.-
.
4
°
•.
.
•
I'
.
;
•
•
.
-
.
•
2
0
-2 in Mvinutes
of Arc
Offer
::
:
-
-20
-20
4
0
0
20
20
Fig. 2.- X-ray Source Identification
Left: offsets between the IPC X-ray sources and the nearest EINLINE point
sources. The "keyhole" contour is the line within which a match is declared; Right:
offsets with Abell clusters with countour lines overlaid. The heavy circle is the
adopted 8 arc-minute matching radius.
Table 1: Point sources with PE > 0.99
ID
01. 2 5
GD 50
9.3
EX0346.2-0106
9.0 AG+52 1002
EX1441.7+5209
10.1 1E1907.0-6405
EX1907.0-6404
6.7 KUV00445-2058
EX0044.4-2058
8.0 1E0536.5-2818
EX0536.5-2817
5.3 QSO1430+626
EX1430.1+6237
3.5 MS1558.2+4123
EX1558.0+4124
132
Object
White Dwarf
F8 star
K4V star
UV source
AGN
AGN
AGN
Offset [']
< 1
<2
<1
< 1
0.4
0.6
1.4
z
N.A.
N.A.
N.A.
N.A.
0.270
0.402
1.168
my
13.77
8.70
11.79
16.30
19.00
18.80
17.69
1
1.2
1.4
1.6
1
1.2
1.4
1.6
1.4
1.6
1.4
1.6
f,/f 1
11111111
1
liii
1.2
1.4
1.6
1
1.2
1.2
1.4
1.6
1
1.2
f 4/f
3
Fig. 3.- Flux ratios of point sources
Ratio of fluxes between apertures 1 (2.5'), 2 (4.7'), 3 (8.4'), and 4 (12.2') for known
points sources with or > 10.
133
30
30
20
20
10
10
'
1
40
1.2
a2/1
1
1.4
1.2
(Ta3/'7
1.4
1
40
I:
30
30
20
20
10
10
~F~j•3ft~jX~c$¼FA4iA
U
1
1.2
9 4/a
0
1.4
4U
30
30
20
20
10
10
1
1.2
1.2
1.4
93/T2
1
40
U
1
U
1.4
1
1.2
1.4
Fig. 4.- Signal-to-noise ratios of point sources
Ratio of signal-to-noise between apertures 1 (2.50'), 2 (4.7'), 3 (8.4'), and 4 (12.2')
for known points sources with a a 1 > 10. The dashed lines are the adopted
extendedness criteria. Extented sources must fall to the right in two or more ratios.
134
SI
I
f
I
I
X
30
20
1.2
1
1.4
1.2
O3/"1U
I I I
1
j
1.2
a
-
I
I
I:
1.4
1
4/0ý1
I
I
I
1.2
(T3/0'
1.4
2
I-
1.2
1
9"4 /u02
1.2
cT
4 /1
1.4
3
Fig. 5.- Signal-to-noise ratios of Abell clusters
Ratio of signal-to-noise between apertures 1 (2.5'), 2 (4.7'), 3 (8.4'), and 4 (12.2')
:for Abell clusters with count rates in all four apertures. The dotted lines are
adopted cut-offs to which objects must fall to the right of to satisfy the
extendedness criterion.
135
0.E
Z
otin
0.2
^ ^^
U.U
^
^^
U.EMSS
^
^^^
1
/NU
EMSS S/N
P
Fig. 6.- Extent Probability & EMSS Recovery
Left: Cumulative fraction of AGN, stars and all IPC sources with probability of
extent < PE. At a threshhold of PE = 0.99, 98.5% of stars and AGN are rejected;
Right: Fraction of EMSS sources that are not recovered in the IPC source catalog
used here vs count-rate.
U
.)1
X
-2
-3
log C.R. [EMSS]
EMSS S/N
Fig. 7.- EMSS/Archival IPC comparison
Left: Comparison between the EMSS reported count rates and the measurements
obtained in the archival IPC source catalog for the 2.5' aperture; Right:
comparison of the signal-to-noise in the IPC sources versus EMSS surveys.
136
0.8
0.6
0.4
0.2
0
z
90%
99%
99.9%
Fig. 8.- X-ray Extent of EMSS Clusters
Probability of extent of EMSS clusters versus their redshift. A histogram of the PE
values is also shown.
137
"
*:
C
4
4
x:
a
P
4
x
it
a
x
n
I
i;;
*I'
*
*
9,:
i
:a
:pi.
~4'
%:
i
4;
4,::
*-t 44
4--·
W .1
e.4
I
99
4it
*
-n
4.*
Fig. 9.-
POSS E images of Abell Clusters
Digitized Sky Survey images from POSS E (red) photographic plates. These are
15 x 15 arc-minute fields of richness class 2 Abell clusters. Clockwise from upper left;
A2256 (z = 0.058), A1235 (z = 0.104), A1094 (z = 0.200), and A1622 (z = 0.286).
138
T
-
a4
"-'
9
r,
+
::::::1
:r
"
c
i
:*
r.:
: r:
,.~
:·
%'-i
r
"
*
_
I
:~
k
:
·
:·
..
- i
;
.
a
:
- -
:r·
is,
I
9~
i:,
:I
1
x
C
11c
J
s
-1. _
-e
a"*
r
r
s
r
*:
i. :C::
:a
rr
r-:
::::
'*
i
-i
:
°
:!Q·
_
r
:a:
--'-r
i
:
::*- 1.1: .
":
:W:
*::: :i :~
-,
Fm
i
..
*-:
"
.
:%
X
:·
;-·
i- I
%
::: :
;h
::X:i*::
-:*
i
*4.
*
#
Fig. 10.- UK Schmidt J Images of Abell Clusters
Digitized Sky Survey images from UK Schmidt J (blue) photographic plates. These
are 15 x 15 arc-minute fields of richness class 2 Abell clusters. Clockwise from upper
left; A3266 (z = 0.059), A2541 (z = 0.108), A2534 (z = 0.198), and S1077 (z =
0.312).
139
100
80
60
640
220
n
4't00
6000
8000
N[4]
.2
.8
.4
Redshift
.8
Fig. 11.- Gunn Filters and Elliptical Colors
Left: Transmittance plot of the Thuan-Gunn gri filter system used in the cluster
galaxy photometry program; Right: Predicted observer-frame g - r colors of
passively evolving elliptical galaxies (Buzzoni 1989). the different curves correspond
to ages of the galaxy when it is observed of 6, 8, 10, 12.5, and 15 Gyr (moving to
redder colors). The points are unrelated observations of clusters.
2
24
. .. . I I
I I'
'
I.
I.. I.I
...
1.5
uý 2
'=
0
.·
+'
r..5
.·r
0
tio
o
•,-
0
·. i
.·.
C
0-
C
1I
'U18
I
20
22
g [g isophote]
,
,
-0.5
,
0.5
24
0
1
0.5
g-r [aperture]
1.5
2
g-r [aperture]
Fig. 12.- CCD Photometry Tests
Left: comparison of g photometry generated from r image isophotes to that from
independent g isophotes. Right: comparison of g photometry obtained with r
isophotes to aperture photometry.
140
EX0806.3+2057 Imaging
Fig. 13.-
•
:.. ..
...
• :: .: ;:- : . : . .
.
. •
. ..
: .
: : 0•
:
..
. .. .. . . .S.
.
at
4 :--ii i-iii:_:~
::
i ii:Z:
4•• •i::; " • :i :
*
a
:·'
*~-~i
iii-: ::
o
00
0
0.
o
0
S
0
o
o
O
0
0
0
0
O
0
00
0
O0o
0
0
0
.0
0
0
o
*O
g-r > 1
I-
I-
0
0
g-r < 1
,, :i
......
......
:.
' . ' I"I..
"...
.
......
.
22
20
-
18
,,
1 6
L
II
0
0.
1.5
Top: 4 x 4 arc-minute section of a 900-second
r
r-band CCD image centered on the X-ray source EX0806.3+2057 (PE = 0.94);
Bottom: color magnitude diagram of galaxies within 2' of the X-ray centroid of
EX0806.3+2057. Middle: spatial distribution of red (g - r > 1) and blue
g - r < 1) galaxies.
141
Table 2: Unidentified Extended X-ray Sources
R.A. [B1950]
hh mm ss.ss
0 18 07.90
0 55 25.77
1 34 48.18
2 24 55.31
2 55 43.32
4 08 58.92
4 10 06.18
4 45 01.94
5 03 35.45
5 26 47.17
5 36 30.52
6 01 09.06
8 37 24.85
8 42 01.67
10 10 21.82
10 45 52.61
11 01 40.51
11 21 08.64
11 43 18.06
11 56 08.83
12 48 37.53
12 49 51.48
12 50 59.90
13 20 14.14
13 22 42.43
13 33 27.49
14 04 28.97
14 30 46.31
14 33 35.50
14 48 41.86
15 06 53.33
15 25 26.52
15 33 51.80
15 43 59.59
15 47 19.05
15 48 19.89
15 59 29.26
16 43 18.11
17 02 18.39
17 06 44.97
17 07 55.45
17 17 29.85
18 52 23.25
Dec.[B1950]
dd mm ss.s
- 75 06 05.7
+ 30 20 17.4
- 05 04 35.9
- 10 41 38.4
+ 20 41 55.7
- 10 21 44.1
+ 07 37 54.6
- 59 37 09.0
- 11 44 09.4
- 20 36 10.2
- 28 17 53.0
- 31 58 52.5
+ 13 12 58.2
+ 19 16 00.0
+ 49 49 03.9
+ 06 59 18.9
+ 45 32 13.3
- 08 35 48.6
- 04 25 56.0
- 27 14 25.8
- 00 56 12.7
- 01 06 58.7
+ 56 35 16.4
+ 42 40 23.3
- 10 56 21.1
+ 51 58 43.1
+ 72 33 24.3
+ 05 53 54.0
+ 19 37 04.2
+ 19 34 20.4
+ 57 13 32.1
+ 11 19 23.4
+ 23 28 27.2
+ 01 00 46.5
+ 05 52 30.6
+ 12 31 11.1
+ 08 39 10.6
- 03 21 48.1
+ 60 55 21.6
+ 49 08 20.0
+ 70 54 46.3
+ 17 39 58.5
+ 79 34 40.8
f' 2
1.848
0.616
3.696
1.540
0.000
2.772
1.232
1.540
0.616
0.924
2.156
0.000
0.924
0.924
3.080
2.772
0.616
1.232
2.772
1.848
1.232
0.616
0.000
0.924
2.772
0.000
0.000
1.540
0.000
3.080
0.000
0.000
0.616
0.000
0.000
0.616
1.540
0.924
1.540
3.388
0.308
1.232
1.232
f2.35
3.080
3.696
5.236
3.080
4.312
3.696
3.080
2.772
1.540
1.848
3.080
5.852
0.924
1.848
4.620
7.084
2.156
3.696
3.696
3.696
2.464
2.156
2.156
3.696
4.312
3.080
3.388
1.848
4.620
6.160
3.696
4.312
3.388
4.620
7.700
1.540
2.772
2.464
6.160
4.928
1.232
1.232
1.540
f 20
4.312
1.848
12.628
7.084
4.312
9.856
5.852
5.852
3.080
3.388
5.852
10.164
4.928
4.928
4.928
11.704
4.620
7.084
8.316
5.544
4.928
3.696
4.928
5.852
63.448
12.320
8.316
5.544
7.084
6.776
12.936
8.932
4.620
14.784
11.396
4.620
7.700
4.620
7.700
12.628
1.848
9.548
4.620
f610
8.316
5.236
19.404
8.932
10.164
13.860
11.704
10.164
6.468
7.392
8.008
19.404
6.160
5.544
13.552
14.784
7.700
10.780
20.020
12.320
6.776
7.084
8.624
10.164
16.632
19.096
15.708
8.932
16.632
16.324
17.864
11.704
4.620
17.864
22.792
9.240
17.556
7.700
5.236
18.172
4.004
9.240
7.700
25
2.75
2.63
3.05
3.61
0.00
2.97
2.69
2.51
3.29
2.73
7.95
0.00
3.46
2.93
3.35
2.57
2.68
3.34
3.53
2.93
3.11
3.15
0.00
2.73
2.83
0.00
0.00
2.93
0.00
3.02
0.00
0.00
2.88
0.00
0.00
2.98
3.58
2.87
2.57
2.90
3.48
3.28
3.03
2. 3 5
2.89
2.58
3.22
3.39
3.09
2.59
2.89
2.68
2.53
2.95
6.37
2.64
3.11
2.78
3.56
2.88
3.53
3.42
3.72
2.52
4.33
2.83
2.70
2.95
3.03
2.69
3.25
2.54
2.94
3.14
2.62
2.66
4.79
3.11
3.18
2.54
3.09
2.55
3.11
3.20
3.54
2.76
3.25
4.20
5.78
3.68
4.58
5.91
3.03
4.68
4.35
4.11
4.74
3.74
8.22
3.67
6.22
4.56
3.16
4.75
3.31
4.40
3.32
4.60
5.91
3.96
2.77
3.21
19.63
3.99
3.72
4.43
3.99
3.26
4.90
3.93
6.40
5.69
3.41
3.61
5.00
4.63
3.55
4.73
4.91
10.39
3.53
a6.10
6.35
6.35
6.53
6.83
5.51
5.76
5.97
6.25
5.58
6.06
10.52
5.50
6.68
5.15
6.34
5.76
6.67
6.11
7.49
6.31
6.97
5.93
5.10
5.57
4.39
6.35
6.62
5.85
5.94
5.81
5.99
5.18
6.01
6.30
6.27
6.78
6.98
5.47
8.59
6.41
7.67
7.30
7.09
Note. - (a) preliminary fluxes calculated assuming a bremstrahllung specctrum, Te = 6keV,
z = 0.25, and NH = 3 x 1020.
142
Chapter 5
Look-Back Time
5.1
A PICTURE-PUZZLE OF GALAXIES AND CLUSTERS
Near a small village in the south-west of France there are the beautiful structures of a
partially-rebuilt Roman villa approximately 1700 years old. The prize heirloom of the site
is a spectacular floor consisting of many tens of thousands of individual color tiles. Along
with the villa ruins the floor, I was told, had lain buried in a farmer's field, and had to be
painstakingly reconstructed after the effects of many years of tilling. May the cosmologists
have it so easy....
However, the results presented in this thesis join a host of other evidence that do
encourage hope for the future decription of the formation and evolution of galaxies. First
of all, it seems clear that the luminosity function of bright galaxies in clusters is described
in basic form by a Schechter function with a- - 1 that is very much like that observed in
the field (Loveday et al. 1992; Marzke et al. 1994; Lin et al. 1996). There is no evidence
for a steep faint-end slope except perhaps at absolute magnitudes where deeper imaging
and a more careful completeness correction is needed. The flat faint-end slope found in the
cumulative LF, and the large size and more-or-less random selection of the cluster sample
provide strong evidence that a steep up-turn in the numbers of galaxies brighter than than
MR < -16.5 (about M* + 4) is very rare in clusters, at least in cluster cores.
It is not altogether surprising that there is an absence of dwarf galaxies in the cores
of clusters, even if they are/were abundant in the field, given that some theories predict a
hostile environment in the former. Dwarf galaxies might be expected to be depleted of gas or
143
destroyed from the effects of stripping or tidal disruption. De Propis et. al. (1995) suggest
that dwarf galaxies remain abundant in clusters because the dense intracluster medium
inhibits the escape of gas from their shallow potential wells; presumably an analogous
population in the field have suffered such a fate. This seems implausible given that the
velocity dispersion of the cluster is so much higher than the escape velocity from the shallow
potential wells of these objects. Another possibility is that the dwarfs have been cannibalized
or tidally disrupted by bright galaxies in the field, but have escaped this fate in the cluster
due to the relatively high velocity dispersion. Finally, dwarf galaxies might be produced in
clusters from the debris of galaxy-galaxy collisions (Moore et al. 1996). My results do not
appear to look favorably on these models.
In addition, although the cores of clusters may be depleted in dwarf galaxies due
to gas stripping or tidal disruption, I note that one is observing the total projection onto
the sky, which includes the outer parts of a cluster as well (a 1 - 2/7r or 36% contribution,
in the spherical cluster model). Thus the data seems to rule out any bright upturn in the
outer parts of the cluster as well.
What is disturbing about the entire situation is the fact that for imaging of clusters
at z = 0.1 - 0.2 the background field counts become the dominant component right at the
absolute magnitude MR - -19 where the purported upturn appears. As I have discussed,
systematic errors in field counts can have devastating consequences for the accuracy of the
LF at fainter magnitudes. Unfortunately, no clusters in my sample have been observed and
analyzed by other researchers, so a direct test case cannot be done. Thus, to re-phrase a
famous astronomer, "the luminosity function wars continue", only now with a little more
ammunition.
The flat luminosity functions found both in the field and in clusters strongly support
the theory that galaxy formation was regulated by the pre-heating of the inter-galactic
medium at some earlier epoch [see §1.1 and Appendix A]. The upturn at faint magnitudes,
if it exists, may be a distinct population of objects whose formation was controlled by other
processes (such as SN heating of the gas).
The small dispersion of the asymptotic magnitudes of brightest cluster galaxies
about a secular evolution with redshift is particularly interesting in light of the emphasis of
their use as "standard candles". This data supports the view that the BCGs evolve with the
cluster and once objects of this magnitude form, they evolve only passively with the intrinsic
144
stellar evolution. The correlation between the morphology/core density of clusters and the
magnitude of the brightest galaxy (at least in the CCD imaging) is very good evidence
that these creatures form during and after a cluster forms. Why there is a gap of about
0.5 magnitude between the first and second-ranked galaxies in clusters is not certain, but a
clue might be found in the case of Abell 2142, a "bimodal" cluster with two giant elliptical
galaxies of nearly equal luminosity. The photometry of Abell 2142 is not secure and so was
not included with the data, but if one plots the absolute magnitudes of the twin BCGs they
fall neatly into the gap. The final evolutionary state of most BCGs may be the coallescing
of two roughly equal systems. The gap indicates a paucity in the number of binary systems,
which assuming that there is no hidden bias in my sample, would indicate that this process
happens in a small fraction of a Hubble time. In my sample there are 4 clusters with binary
Rood-Sastry types and 50 cluster with less-evolved Rood-Sastry types, suggesting that the
mergin time is les than a tenth of a Hubble time (-
1 Gyr). The relative luminosity of
the brightest galaxy fainter than the gap seems to be a good evolutionary "clock" for the
cluster. The appearence of the BCG at MR- - 23.7, however, signifies that the clock has
stopped (and in fact is slowly ticking backwards).
My observation of an increase in the number of disk-like systems in clusters with
redshifts does agree with the expected Butcher-Oemler effect and does suggest evolution
of galaxy morphology in clusters, where disks are being turned into earlier types. Finer
discrimination using colors and a larger sample to a higher redshift is needed to determine
if the effect does indeed parallel that of Butcher & Oemler (1984), and whether spirals are
being turned into SO galaxies or ellipticals. This last issue is a very important distinction,
since if disks are turned into SOs in clusters (as seems plausible), the formation of elliptical
galaxies must have been a separate, more primordial process. Perhaps both "nature" and
"nurture" are responsible for these types of galaxies.
Finally, my search of the archival Einstein IPC database has turned up a large
number of cluster candidates, some of which, like EXO806.3+2057, which appear to be
distant clusters in follow-up multicolor CCD imaging. What I do not find is any evidence
for a large number of bright, nearby X-ray clusters much in excess of the Abell catalog.
Thus the discrepancy between the normalization and evolution in optically-selected and
X-ray selected clusters remains a mystery.
145
5.2
LESSONS LEARNED FROM
MY FIELD WORK
Although interesting results have been obtained from this data, a considerably great deal
more could be learned by a more careful and systematic effort that uses some of the lessons
learned from the many mistakes made here.
In terms of luminosity function construction, the choice of cluster sample certainly
deserves more careful attention so that a uniform sample is created that covers the full
range of some parameter of interest, such as the dynamical age of the clusters. In particular,
observing clusters in some narrow range range of redshifts (perhaps z - 0.15) now seems
like a very good idea. Selecting clusters from the Rosat Brightest Cluster Sample (Ebeling
et al. 1996), or Brightest Abell Cluster Sample (Jones et al. 1996) (which were not available
at the beginning of this project) is an obvious thing to do. Another possibility are the
nearby, bright X-ray clusters for which Rosat pointed imaging with long integration times
is available.
The X-ray surface brightness morphologies can be used as an indication of
the dynamical state of the cluster, and hence its age (Buote & Tsai 1996). An alternative
scheme would be to select clusters with photometry of the first-ranked galaxies and use
the hypothesis, supported by the results of this thesis, that these objects grow to be the
Brightest Cluster Galaxies (or "Asymptotic Cluster Galaxies"), and until reaching that
limit serve as a "clock" for the host cluster.
In a more technical vein, the plethora of possible systematic errors in background
subtraction, many of which were discovered as the project progressed, strongly suggest the
use of a differential background subtraction rather than unassociated fields. Perhaps the
optimal strategy would be to image in a strip running from the cluster core out to a distance
of about 3 Mpc where the cluster density has fallen to a very low value. The "drift scan"
mode of CCD operation in which the telescope is fixed and the CCD clocked to match the
sidereal rotation rate of the Earth, comes to mind. For cluster out to z - 0.2 one should
exploid the Digitized Sky Survey to center the imaging and avoid the pitfall of the erroneous
Abell coordinates. The use of filters in common with the large sky surveys to avoid the
uncertainties associated with transforming from one photometry system to another is very
important: It seems clear that to compare with the Las Campanas Redshift Survey and
the upcoming Sloan Digital Sky Survey, the Gunn system filters are the pass-bands of
choice. Finally, the amount of additional science that can be obtained with the use of an
146
additional filter to construct colors is well worth the extra observing time required. The
color information is also necessary for more accurate estimates of the K correction!
5.3
LOOKING FORWARDS
One of the more interesting and timely topics explored in this thesis has been the development of the "Q" parameter for the morphological classification of galaxies. It seems to
separate disk systems and spheroidal systems with accuracy and has been demonstrated to
be a quantity that is quite robust to large variations in seeing and signal-to-noise. A potentially powerful technique would be to combine it with a measurement of the mean surface
brightness and a single color to attempt to resolve out differences in morphologies, stellar
populations, and star-formation rates in distant galaxies. Application of this classification
to Hubble Space Telescope images of moderate depth will be useful. Another application
is to the low signal-to-noise images generated in obtaining CCD photometry for redshift
surveys such as the Las Campanas Redshift Survey.
The other topic of great interest is the passive evolution of the elliptical population in
clusters. Mounting evidence, e.g., (Stanford et al. 1995) points to absolutely very little going
on with these objects other than the normal evolution of a stellar population that formed
at very high redshift. Multi-color photometry of these objects, combined with the latest
stellar evolution and spectral synthesis codes, can begin to place some interesting limits on
the redshift of formation (Rakos & Schombert 1995). This problem ties in directly with
the now well-known conflict between the best estimates of the Hubble constant (60-75 km
sec - 1 Mpc - 1 ) and the estimated ages of globular clusters (- 14 Gyr). It would be as if, as
paleontologiests, we discovered fossils that were older than the Earth! These values cannot
be reconciled in standard cosmologies without resorting to the Faustian bargain of invoking a
cosmological constant. Brave souls have already begun to explore the possibility of testing
cosmological models at more than one epoch (i.e., now) by using the color evolution of
cluster ellipticals (Buzzoni et al. 1993). More fundamental (and less controversial) tests
of world models have also been done (Pahre et al. 1996) and the possiblity of using the
basic properties of elliptical galaxies as yardsticks to measure the decelleration parameter
has been proposed. Of course, much of this is entangled with our knowledge of stellar
evolution and the formation of stars in those ancient starbursts, but the size, homogeneity
147
and well defined behavior of the sample does lead one to dare hope to know the cosmological
parameters at last.
Will we see a time when the formation of galaxies and subsequent evolution of galaxies is well understood, with studies confined to the exceptions rather than the rule? It may
seem a ridiculous possibility but, staring at the image of the Hubble Deep Field (Williams
et al. 1995) which probably sees through the entirety of the optically observable universe,
the question does not seem so silly. Will the day arrive when tests using distant objects
like elliptical galaxies, among other tests, finally converge to some irrefutable cosmological
model whose parameters, be they Q, A, or whatever, are quickly measured to an accuracy sufficient for astronomers, and, much more slowly, to a precision acceptable to atomic
physicists; set to take their place alongside the speed of light and Planck's constant in the
"CRC"? On a personal note, this author hopes the answer is no: May there always be that
fainter, more distant object to see, that piece of the puzzle that does not fit, that artifact
that does not make sense, that bone that does not belong, and always one more Mask that
hides the Face.
148
Appendix A
Eine Kleine Theory: Galaxy Formation
In the succeeding discussion I will often assume Q = 1. Although there is a body of
observational evidence suggesting that Qm < 1, perhaps taking a value - 0.2) (Coles &
Ellis 1994), there is also substantial evidence that the bulk of the galaxy formation occured
at a redshifts of 3 or higher (Steidel et al. 1994; Steidel et al. 1996) where Qm would
have been much closer to unity . In a matter-dominated cosmology with zero cosmological
constant the evolution of Qm is approximated as 1 - (1 - Qo)(1 + z)- 1 . Qrm is even closer
to unity in a universe with non-zero vacuum energy (cosmological constant).
Modern observational cosmology is based on the premise that the object observed on
galactic scales and larger are the products of the evolution of primordial perturbations in an
otherwise extremely homogenous early universe (Peebles 1993). The first ingredient in any
galaxy formation recipe is the description of these fluctuations in mass density. They are
usually assumed to be random and Gaussian-distributed with a power-spectrum described
by a power-law over the range of mass scales of interest. After the mass and radiation fields
decouple at a redshift of z-1400 density enhancements begin to grow under the influence
of the corresponding gravitational potential.
A second ingredient is a model of the growth of these fluctuations: For the mass
scales of interest here this evolution has extended well into the non-linear regime: galaxies
and clusters of galaxies are - 104 and - 102 times denser than the mean matter background.
An analytic treatment of the evolution of non-linear, gravitationally bound objects was
attempted by Press & Schechter (Press & Schechter 1974), who introduced an ingenious
description of the mass distribution of gravitationally-bound objects as they evolve to larger
149
and larger scales from an intitial mass perturbation spectrum. The fractional root-meansquare fluctuations in the mean matter density on the mass scale M are assumed to obey
the scaling law
)2112
M-a,
(A.1)
where the value of a is related to the index of the power spectrum of the initial density
perturbations by a = (3 + n)/6. They found the number distribution of bound masses at a
particular epoch to obey the law
n(M) = n0o YM)l"
2f
(A.2)
[note that the a defined here is 1 - a in Press & Schechter (1974)]. This distribution is
essentially a power-law which is exponentially truncated at a characteristic mass M*. In
an Einstein-de Sitter universe with the matter density equal to the critical value (Qm = 1)
the value of M* evolves with redshift as
M* = MO(1++ z) 1/
(A.3)
The Press-Schechter description has been compared with more sophisticated analytical descriptions (Bond et al. 1991) and numerical n-body simulations and has held up remarkably
well (Navarro et al. 1996).
The simplest model for an individual collapsed object is the spherical collapse model
for dissipationless matter. The objects collapses and virializes until its mean density is a
factor of 178Q-0' 6 times the mean density at formation (for a flat universe) or 178Q - 1 for
an open universe. If Qm1 and structure formation proceeds hierachically then objects will
survive only on the order of a dynamical time before becoming part of a larger object and
the mean density at formation is the density at that particular epoch, i.e. there are no
"old" halos. The internal structure of the halo is taken to be an isothermal sphere with
density profile p - r - 2 . With these assumptions the relation between the circular velocity
and mass of the halo is (White & Narayan 1987)
3
M
V
9GH
9GH
150
(A.4)
where H is the Hubble constant at the formation epoch. A primordial gas component (76%
H and 24% He) in hydrostatic equilibrium with this potential well will have a temperature
(nearly independent of Q) of
Thalo-4.4 x 106 h2/ 3 (1 + z)
(
'102M)
K.
(A.5)
The dynamical time (also roughly the crossing time or collapse time) of a homogeneous
sphere of density p is (Binney & Tremaine 1987),
td
S16Gp
(A.6)
With the previous assumptions, the dynamical time in Gyr is
td0O.18-'/2h- (1 + z)- 3/2
(A.7)
The final ingredient to galaxy formation is the introduction of gas into the evolving potential
wells such that it can eventually contract to a dense state and form stars. The physical
processes associated with the gas will also introduce length scales into an otherwise nearly
scale-free model.
It was suggested that the particular mass scales of objects (globular
clusters, galaxies, clusters) might arise from just such physics (Press & Schechter 1974). It
was first pointed out by Peebles & Dicke (1968) that the typical mass of globular clusters,
some of which appear to be the oldest objects in the universe, is close to the Jeans instability
mass at the temperature at the epoch of decoupling (- 104"K) where the matter was first
able to move independently of the radiation field.
Several researchers (Binney 1977; Rees & Ostriker 1977; Silk 1977) independently
developed as a basic criterion for galaxy formation the requirement that the cooling time
scale of the gas must be shorter than the dynamical time of the object, allowing the gas to
cool faster than the collapse of objects can heat it to the virial temperature. Its entropy
reduced, the gas collapses to a dense state which promotes efficient star formation and
a galaxy is born. White & Rees (1978) combined all three ingredients, proposing that a
"dark" dissipationless matter component dominated the evolving gravitational potential on
the mass scales of galaxies and collapsed into bound objects from an initially perturbed
state. The dissipative (gas) component falls into the potential wells and is also influenced
151
by external (ionizing flux) and internal (star formation) heating sources while it is also able
to radiate away energy. Galaxies form when the gas is able to radiate away a large amount
of the gravitational energy and collapse to a very dense state to form stars. The resulting
objects have undergone significant dissipation and are thus much less susceptible to further
evolution by merging. This process essentially "freezes out" the structure at a particular
epoch and makes it visible as the distribution of galaxies observed today, malgre some mild
evolutionary effects. Subsequent work has been refinement of this basic picture.
The formation of luminous objects is first allowed after recombination at a redshift of
z
-
1200, although any non-baryonic "dark" matter ingredient could start forming structure
earlier. Even at somewhat later epochs any ionized portion of the gas is strongly coupled
to the cosmic radiation background and transfers a drag force to the bulk of the gas and
prevents it from freely collapsing into the gravitational potentials of bound objects. The
"Compton drag" force per unit volume on a gas with ion density pi moving at velocity - is
16picaTUBTv4
F3mpC2
(A.8)
where the black-body temperature of the radiation field is T = 2.731(1 + z). The gravitational volume force experienced by the gas falling into a dark-matter halo is roughly,
FT = PVc/td
(A.9)
where v, is the circular velocity of the halo (which will be the mean free-fall speed of the
gas) and td is the dynamical time. Equating these two expressions leads to an expression
for the maximum redshift at which Compton drag permits baryonic matter to collapse into
form collapse objects:
-1
1 + zc,,366( /1h2/55f
(A.10)
where fi is the ionization fraction of the gas.
Once Compton drag disappears, the gas must be able to radiate away a significant
amount of its binding energy in the dark-matter halo. The primary mechanisms by which
gas cooled in the early universe are cooling by ions scattering cosmic background radiation
photons (Compton cooling), ion-ion scattering (bremstrahllung), and recomination radiation. The ratio of self-radiation to Compton cooling for collapsed objects while Qm
152
1
is 3300(1 + z)-'QBh 2T6-1 . For QB = 0.05 and h = 0.7 radiation cooling dominates below
z , 80 in the formation of galaxy-sized objects.
In the temperature range of primary interest (104 to 105 K) the dominant cooling
mechanism of primordial gas (76% H and 24% He) is radiation from collisionally excited
partially ionized atoms (HI and Hell) (Fall & Rees 1985).
Below 15000 K the cooling
efficiency drops off precipitously as the gas rapidly becomes neutral. Thus luminous objects
may not form unless the equivalent temperature of dark-matter halos exceeds 1.5 x 1040 K.
The temperature of a dark-matter halo, independent of cosmology, is given by
T = 2.3 x 106(1 + z)
01(fL)
(A.11)
where for convenience I have substituted luminosity L for mass, assuming a constant massto-light ratio of 50. Thus the smallest objects which can form at the end of the Compton
drag-inhibited epoch have L - 1000L®.
If we live in an Q = 1 universe with spectral index n and if objects of luminosity
4 x 101 1h- 2 L
are now reaching the non-linear phase of collapse then collapsing objects at
redshift z will have luminosity
L = 4 x 1011(1+
z) -
6/(n+ 3)
(A.12)
For n < -1 all objects forming after Compton drag becomes negligible have temperatures
above 1.5 x 1040 K and are thus able to cool efficiently.
The ratio of the cooling time to the dynamical time for T - 106 K is
t~0.9h-1/3 B
1/2(
+
B
td
)-1/2
4 x-012
2/3
12
x10
)
(A.13)
Equating this to unity, the maximum mass of a galaxy forming at redshift z is
M
,
1 / 2QQ-1/2(1 + z) 3 / 4 M®
5 x 10 12 hl/2
(A.14)
which, for the usual choice of parameters and a formation redshift of 3, gives 1.3 x 1012 M®.
Thus only objects less massive than Mc will form galaxies, with
Mc , 3 x 1011(1 + z) 3 /4M 0
153
(A.15)
The critical halo temperature below which luminous objects can form is
T,-4.8 x 106 SB2-1 / 2 h(1+ z)3/2K
(A.16)
For 2B = 0.05, Q = 0.2, and h = 0.7, bright galaxies T - 2 x 106 K can form only to z , 2.
Galaxy formation is more efficient at higher redshift because of the higher densities
leading to greater cooling. The objects forming at the present epoch (clusters of galaxies)
have cooling times much larger than their dynamical times, and in most cases, comparable
to the Hubble time.
Perhaps the simplest method of constructing a theoretical galaxy luminosity function
while retaining some basic physics is one which combines the evolution of dark matter halos
described by the Press-Schechter formalism with the requirement that t, < td. Although the
dark matter halos continue to grow in mass, the luminous components of the galaxies that
form are assumed to resist further merging due to their more compact dynamical states.
However, the ability of low mass objects to cool efficiently at high redshift would mean that
most of the gas would reside in very low-mass objects forming as soon as Compton drag
became small. This would mean that the characteristic mass would be of order 106 -10'M®;
i.e., objects intermediate those of globular clusters and galaxies. The characteristic sizes of
galaxies at the present epoch are three to five orders of magnitude larger. This is sometimes
called the "Cooling Catastrophe".
Four possible ways to save this simple picture of galaxy formation are (1) adopt
a power spectrum index that is much steeper than the CDM value of -1 to form more
massive halos at higher redshift; (2) allow the luminous material continued to merge within
the dark matter halos under the influence of dynamical friction; (3) include the feedback
effects of star formation on the gas such that gas is lost from the shallow potential wells of
dwarf galaxies and they form inefficiently; or (4) that injection of entropy from an initial
generation of stars prevented the gas from collapsing and cooling until the halos grew to a
critical mass. I next address each of these avenues in order:
To form L. objects at a redshift of 100, even in an open universe, would require a
spectral index of at least n = 3, absolutely inconsistent with the available data on these
scales; (White 1993) and references therein.
It is straight-forward to show that merging of luminous material would not have
154
been able to form L* galaxies.
The orbital decay time of globular clusters within the
isothermal sphere (p , r - 2) potential of an idealized dark matter halo has been worked out
by Tremaine (1976):
td
)
10 10M
.9
10 6M100
yr,
(A.17)
where M is the mass of the galaxy, and m and ri are the mass and initial orbital radius of
the globular cluster. The merging time or dynamical time for the halo can be written in
the same style:
tmo2.1 x 10
2kpc
0
1010Me)
yr
(A.18)
For an Q = 1 CDM universe the redshift dependence of the dynamical friction time-scale
is (1 + z) - 9/ 2 , while that of the halo merging time scale is (1 + z) - 3/2. Thus at an early
epoch globular cluster-mass objects could coallesce at a rate faster than the merging of halos
could disrupt them, but eventually dynamical friction became very inefficient and galaxy
formation by this mechanism halted. The halo mass scale at which this occurs is found by
simply equation td and tm to find
Mrit~1.8 x 108 M®
(A.19)
Note that tidal stripping of the globular cluster will increase the orbital decay time and
thus decrease the derived value of the critical mass.
White & Rees (1978) developed a model which includes the self-regulating effect
of supernovae energy input during galaxy formation and supposes that the mass of gas
turned into stars is proportional to the specific binding energy of the galaxy, which in
turn is proportional to the circular velocity squared. With an appropriate adjustment of
parameters, the mass of a typical galaxy can be produced by this model although the
faint-end slope is found to be (n - 13)/(n + 7), or -2 for CDM, steeper than observed.
An alternative model, developed by Blanchard et. al. (1992), which invokes similar
physics postulates an injection of entropy into the IGM sufficient to prevent formation of
luminous objects until the gas has cooled and dark matter halos grown large enough to
gravitational trap it. Such an episode of "reheating" is consistent with the high level of
ionization required to avoid the Gunn-Peterson effect and may have been produced by an
155
initial generation of stars (Population III). The gas would not fall into a potential well and
form a galaxy until the condition Tg < Th was satisfied.
The ratio of Compton to radiation cooling in a uniform IGM at temperatures above
1
oK) /2
10oK (where free-free emission dominates) is given by 0.2(1 + z)h-2Q (4.4e
and thus for reasonable values of the cosmological parameters, Compton cooling dominates
to the present epoch. In a flat, A = 0 cosmology the ratio of the Compton cooling time
to the Hubble time (at redshift z) is 120h(1 + z) - 5/ 2 and the IGM above a redshift of - 6
cools by Comptonization of CBR photons, while below this redshift it cools adiabatically.
The redshift evolution of the temperature in the latter case (the only one of interest here
as I shall show) is _ (1+ z) 2 .
The velocity dispersion, expressed in terms of temperature scales as
T, , (1 + z)M
2/ 3
(A.20)
independent of the cosmology. Since the temperature of the IGM scales as (1 + z) 2 , the
critical halo mass for galaxy formation would scale as (1 + z) 3 /2 At the epoch that galaxies
can form in halos of mass M, the characteristic mass is M.(0)M-4/(n+3 ). Convolving this
with the result of Press & Schechter (1974), the result is mass distribution of the form;
n-5
n(M)dM , (M/M,)M
/
e-(M/M.)
n+7
(A.21)
Interestingly, an n = -1 choice of spectral index reproduces the flat luminosity function
observed in field surveys. Furthermore, this model predicts that the most massive galaxies
formed first. These would have the lowest specific angular momentum and would thus tend
to be early-type, as is observed.
Sufficiently sensitive measurements of the spectrum of the cosmic microwave background (CMB) can be used to test the early heating theory (the late heating theory is more
difficult to test since the gas is clumped and occupies a much smaller net solid angle.) The
distortion of the CMB away from a perfect thermal spectrum by a hot IGM is given by the
y parameter;
S
TneckTe
dt.
mec 2
(A.22)
If the IGM is heated to some initial temperature Ti at redshift zi and cools adiabatically
156
thereafter until recombination at z, then, for an Einstein-de Sitter Cosmology,
y-3.4 x 10-
6
Bh
Ti
106 0oK
(1 + Z)
3
/2
(A.23)
If L. galaxies (with T , 5 x 106 0K) were able to form at a redshift z* then, for the current
limit of y < 2.5 x 10-5 (Wright et al. 1994), we have
1 + zi < 1.1(1 + z*) 4 / 7 (sBh)
- 2 7
/ .
(A.24)
For z* = 3, OQ = 0.05, and h = 0.7, zi < 5.3.
The most massive galaxies have an X-ray temperature of - 1 keV. These could
not have formed before a redshift of 5. The stringent limits placed on formation epochs in
this scenario means that more precise measurements of the y parameter will prove to be a
decisive test of the model.
Important here, there are two observational tests of these formation theories having
to do with the galaxy luminosity function. The first is to measure the slope of the faint
end of the luminosity function. A steep slope (e.g. a = -2)
would support the picture
that dwarf galaxies formed first (albeit inefficiently) while a flat slope (a = -1) would
support a bright galaxy-first scenario. A second test would measure the dependence of the
characteristic value of M* on the local matter density. If bright galaxies formed later they
would be systematically brighter in areas of high overdensity such as clusters of galaxies.
Any correlation would be much weaker in a picture where the bright galaxies formed first.
157
Appendix B
Cluster Luminosity Functions
B.1
PHOTOMETRIC LUMINOSITY FUNCTIONS
Luminosity functions constructed from photometric observations of 24 Abell clusters. Points
are completeness-corrected data binned in 1-magnitude bins. The curves are the best-fit
Schechter functions with both M* and a allowed to vary. Error bars are the Poisson noise.
158
Abell 1632
Abell 1667
Abell 1675
,,,,
21.5
:
:
:
2
-1.5
1
-.
1
0.5
-
0.5
0.5
0
0
0 -22 -20
-18
-16
-14
1
Abell 1679
I
Abell 1793
Abell 1878
II I I I I
4.D
tI
I
fiIl I
I
II
I
2
1.5-
0
1
1
0.5
0.5
0
-22 -20
Abell 1889
Abell 1929
-18
-16
- 14
-16
-14
Abell 1952
2
1.5
1
0.
0.5
0.5
0
-22
-20
159
-18
-16
-14
-22
-20
-18
Abell 2056
Abell 2061
Abell 2069
Abell 2083
Abell 2084
Abell 2142
Abell 2142
Abell 2192
Abell 2198
,,
2
1.5
1
0.5
0
160
-22
-20
-18
-16
-14
Abell 2211
Abell 2213
Abell 2235
Abell 1793
Abell 2266
2
I-
A/
i-i
F- I-22
-i
I-20 -18
-16
-
Abell 2244
2.5
2.5
2
2
1.5 -
1.5
1
1
0.5
0.5
-22 -20 -18
-16
-14
-22 -20 -18
161
-16
-14
B.2
NON-PHOTOMETRIC LUMINOSITY FUNCTIONS
Luminosity function of the 43 Abell clusters observed under non-photometric conitions.
The histograms are the LFs constructed by separating galaxies inside and outside an aperture centered on the peak of the cluster surface brightness distribution. If the cluster is
sufficiently distant, a radius of 0.5 h - 1 Mpc is used; otherwise, the radius is decreased by
10% increments until the solid angle outside of the aperture is at least 1/3 of the solid angle
inside.
162
Abell 1
Abell 7
Abell 24
Abell 31
Abell 41
Abell 84
Abell 96
Abell 136
Abell 175
163
--e
Abell 1
Abell 7
Abell 24
Abell 31
Abell 41
Abell 84
Abell 96
Abell 136
Abell 175
-
U -I--
-1b
-14
-22
-20
164
-18
-16
-14
-22
-20
-18
-16
-14
Abell 1
Abell 7
Abell 24
Abell 31
Abell 41
Abell 84
Abell 96
Abell 136
2
1.5
1
0.5
n
-di
-22
2
-20
-18
-16
-14
r.
0.
Abell 175
€• •
2
1.5
1
0.5
n
-22
165
-20
-18
-16
-14
Abell 1
Abell 7
Abell 24
Abell 31
Abell 41
Abell 84
Abell 96
Abell 136
Abell 175
166
Abell 1
Abell 7
Abell 24
,,
2.5
2
2
1.5
1.5
1
1
0.5
0.5
-22
-20 -18
-16
0
-14
Abell 31
Abell 41
Abell 84
,,
2.5
2
1.5
1
0.5
n
V -22
Abell 96
Abell 136
-20
-18
-16
-14
Abell 175
,,
ne
2
2
1.5
1.5
1
1
0.5
0.5
0
v
.
-22
.
-20
.
-18
.
-16
.
-14
S
-
-to
167
-la
-10
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v
.
-22
.
-20
.
-18
.
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.
-14
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